DiscretizingNonlinearFunctions.html

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<h1 id="Discretizing-Nonlinear-Functions-for-Optimization">Discretizing Nonlinear Functions for Optimization<a class="anchor-link" href="#Discretizing-Nonlinear-Functions-for-Optimization">&#182;</a></h1>
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<p><b>Connor Owen</b>
<br><a href="mailto:cowen33@gatech.edu">cowen33@gatech.edu</a></br></p>

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<p>
One important assumption for any linear program, or mixed integer linear program, is that all functions in the program be <i>linear</i>. While there are techniques to solve nonlinear programs, they generally are computationally more expensive, and often intractable (i.e., not practically solvable). Instead of using these nonlinear techniques, there are ways that we can approximate nonlinear functions with linear functions so that we can use typical linear optimization engines. 
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In this notebook, we'll cover a simple formulation for approximating a nonlinear function in a linear program in one dimension. After, we'll cover an improvement to the formulation that requires some simple modifications. 
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We'll be using the libraries <a href="http://www.numpy.org/">NumPy</a> and <a href="https://matplotlib.org/">Matplotlib</a> for math and graphing functions, and then <a href="https://pythonhosted.org/PuLP/">PuLP</a> for formulating and solving the optimization problems. 
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<h2 id="Discretizing-a-Nonlinear-Function-in-One-Dimension">Discretizing a Nonlinear Function in One Dimension<a class="anchor-link" href="#Discretizing-a-Nonlinear-Function-in-One-Dimension">&#182;</a></h2>
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<p>To illustrate the formulation, we will use a toy problem of optimizing the function $ x \sin{(\frac{8 \pi x} {10})} $ over the domain $[0, 10]$. Below we graph the function using Matplotlib.</p>

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<div class="prompt input_prompt">In&nbsp;[7]:</div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>

<span class="k">def</span> <span class="nf">sinFunction</span><span class="p">():</span>
    <span class="n">Fs</span> <span class="o">=</span> <span class="mi">10</span>
    <span class="n">f</span> <span class="o">=</span> <span class="mi">4</span>
    <span class="n">sample</span> <span class="o">=</span> <span class="mi">10</span> <span class="c1">#end of domain</span>
    <span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">sample</span><span class="p">,</span> <span class="mf">0.01</span><span class="p">)</span> <span class="c1">##step size of 0.01 to make high quality graph</span>
    <span class="n">y</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">f</span> <span class="o">*</span> <span class="n">x</span> <span class="o">/</span> <span class="n">Fs</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s1">&#39;f(x)&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>

<span class="n">sinFunction</span><span class="p">()</span>
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The basic idea behind discretizing this function is that we will create break points to split the domain into several segments, take the function value at the breakpoints of the domain, and then connect these points with straight lines. The resulting piecewise line segment will be a linear approximation of our nonlinear sin function.
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Below, we will define a function to make a list of the x-values of the breakpoints given a number of segments and the length of the domain. We also will define a function to return the function value of our function above given an x-value.
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">createBreakPoints</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="n">xRange</span><span class="p">):</span>
    <span class="sd">&quot;&quot;&quot;</span>
<span class="sd">    numSegments is the number of segments that we will have after we split up the domain</span>
<span class="sd">    xRange is the length of the domain that we are considering </span>
<span class="sd">    &quot;&quot;&quot;</span>
    <span class="n">breakPointValues</span> <span class="o">=</span> <span class="p">[</span> <span class="n">i</span><span class="o">*</span><span class="n">xRange</span><span class="o">/</span><span class="p">(</span><span class="n">numSegments</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)]</span>
    <span class="k">return</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">)</span>


<span class="k">def</span> <span class="nf">optFunction</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="n">Fs</span> <span class="o">=</span> <span class="mi">10</span>
    <span class="n">f</span> <span class="o">=</span> <span class="mi">4</span>
    <span class="n">functionValue</span> <span class="o">=</span> <span class="n">x</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span> <span class="o">*</span> <span class="n">f</span> <span class="o">*</span> <span class="n">x</span> <span class="o">/</span> <span class="n">Fs</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">functionValue</span>
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<p>Let's take a look at what our approximate function looks like with varying levels of segments.</p>

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<h3 id="12-Segments">12 Segments<a class="anchor-link" href="#12-Segments">&#182;</a></h3>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numSegments</span> <span class="o">=</span> <span class="mi">12</span>
<span class="n">xRange</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">breakPointValues</span> <span class="o">=</span> <span class="n">createBreakPoints</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="n">xRange</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">):</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])],</span> <span class="s1">&#39;ro-&#39;</span><span class="p">)</span>

<span class="n">sinFunction</span><span class="p">()</span>
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"
>
</div>

</div>

</div>
</div>

</div>
<div class="cell border-box-sizing text_cell rendered">
<div class="prompt input_prompt">
</div>
<div class="inner_cell">
<div class="text_cell_render border-box-sizing rendered_html">
<h3 id="25-Segments">25 Segments<a class="anchor-link" href="#25-Segments">&#182;</a></h3>
</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In&nbsp;[15]:</div>
<div class="inner_cell">
    <div class="input_area">
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numSegments</span> <span class="o">=</span> <span class="mi">25</span>
<span class="n">xRange</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">breakPointValues</span> <span class="o">=</span> <span class="n">createBreakPoints</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="n">xRange</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">):</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])],</span> <span class="s1">&#39;ro-&#39;</span><span class="p">)</span>

<span class="n">sinFunction</span><span class="p">()</span>
</pre></div>

</div>
</div>
</div>

<div class="output_wrapper">
<div class="output">


<div class="output_area"><div class="prompt"></div>


<div class="output_png output_subarea ">
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"
>
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</div>
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<div class="prompt input_prompt">
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<div class="text_cell_render border-box-sizing rendered_html">
<h3 id="100-Segments">100 Segments<a class="anchor-link" href="#100-Segments">&#182;</a></h3>
</div>
</div>
</div>
<div class="cell border-box-sizing code_cell rendered">
<div class="input">
<div class="prompt input_prompt">In&nbsp;[16]:</div>
<div class="inner_cell">
    <div class="input_area">
<div class=" highlight hl-ipython3"><pre><span></span><span class="n">numSegments</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">xRange</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">breakPointValues</span> <span class="o">=</span> <span class="n">createBreakPoints</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="n">xRange</span><span class="p">)</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">):</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])],</span> <span class="s1">&#39;ro-&#39;</span><span class="p">)</span>

<span class="n">sinFunction</span><span class="p">()</span>
</pre></div>

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<p>As you can see with the above illustration, the more segments we define, the more accurate an approximation we create of the sin function. In general, the more segments we define, the closer we will get to the optimal value of the nonlinear function.</p>

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<p>Now we will cover the optimization problem that we wish to solve. First, the nonlinear version:</p>
$$\max_x \left\{ x \sin{(\frac{8 \pi x} {10})}: 0 \leq x \leq 10 \right\} $$<p><br></br></p>
<p>
Now, we will begin the discretization of our function. Source [1] shows this implementation as Model II. We begin by dividing the $x$ domain into convex combinations of $k$ breakpoints, since any $x$ value can be written as the convex combination as two consecutive points in the domain. 
</p>$$ x = \sum^{k}_{i \ = \ 0} a_i \xi_i \tag{1}$$$$ \sum^{k}_{i \ = \ 0} \xi_i = 1 \tag{2}$$$$ \xi_i \geq 0 \; \mathrm{for} \; 0 \leq i \leq k \tag{3}$$<p>The constraints above allow us to define the domain as convex combinations of two points. If we assume that the one to two breakpoints are consecutive, then we can define the approximate function value as the convex combination of the breakpoints and the function values where the selected breakpoints are:</p>
$$\widehat{f(x)} =  \sum^{k}_{i \ = \ 0} f(a_i) \ \xi_i \tag{4}$$<p>In order to ensure that any positive breakpoints are consecutive, we create a decision variable $z_i$, which is an indicator variable for the segments between the breakpoints; 1 if the segment is selected, 0 otherwise.</p>
$$ 0 \leq \xi_0 \leq z_0 \tag{5}$$$$ 0 \leq \xi_i \leq z_{i-1} \;\;\;\; \forall 1 \leq i \leq k - 1 \tag{6}$$$$ 0 \leq \xi_k \leq z_{k-1} \tag{7}$$$$ \sum^{k-1}_{i \ = \ 0} z_i = 1 \tag{8}$$$$ z_i \geq 0 \;\;\;\; \forall \ 1 \leq i \leq k - 2 \tag{9}$$<p>Constraint (5) ensures that if the first breakpoint is positive, then the first segment is positive; constraint (7) ensures the same for the last breakpoint and the last segment. The constraints from (6) ensure that for the middle breakpoints, if they are positive, then the segment to its left must be positive.</p>
<p>
Line (8) ensures that exactly one segment is selected, and (9) ensures non-negativity for $z_i$ (but note that $z_0$ and $z_k$ are guaranteed positivity from constraints (5) and (7) respectively). 
</p>
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<h2 id="Modeling-a-Nonlinear-Function-Approximation-with-PuLP">Modeling a Nonlinear Function Approximation with PuLP<a class="anchor-link" href="#Modeling-a-Nonlinear-Function-Approximation-with-PuLP">&#182;</a></h2>
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<p>Now that we have our theoretical model formulation, we can begin the Python formulation with PuLP. We instantiate a problem object, and declare our primary decision variables.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">pulp</span> <span class="k">import</span> <span class="o">*</span> 

<span class="c1">##create the problem variable</span>
<span class="n">prob</span> <span class="o">=</span> <span class="n">LpProblem</span><span class="p">(</span><span class="s1">&#39;Linearization Model&#39;</span><span class="p">,</span> <span class="n">LpMaximize</span><span class="p">)</span>

<span class="c1">##primary decision variable##</span>
<span class="c1">#---------------------------#</span>
<span class="c1">#-domain variable</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">LpContinuous</span><span class="p">)</span>

<span class="c1">#-variable for approximating nonlinear function</span>
<span class="n">approxFunctionValue</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="p">(</span><span class="s2">&quot;w&quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">LpContinuous</span><span class="p">)</span>
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<p>Next we can begin defining the segment variables and the breakpoint variables. When we write our final optimization function, we will make these variables as a function of the number of segments that we want. We also define a list of the breakpoint domain values (as we did above)</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">##-create breakpoint variables</span>
<span class="n">breakpointVars</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="o">.</span><span class="n">dicts</span><span class="p">(</span><span class="s2">&quot;partition var&quot;</span><span class="p">,</span> 
                                <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)),</span>
                                <span class="n">lowBound</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span>
                                <span class="n">cat</span> <span class="o">=</span> <span class="s1">&#39;Continuous&#39;</span><span class="p">)</span>

<span class="c1">##-create segment variables</span>
<span class="n">segmentVars</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="o">.</span><span class="n">dicts</span><span class="p">(</span><span class="s2">&quot;segment var&quot;</span><span class="p">,</span>
                                <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">)),</span>
                                <span class="n">lowBound</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span>
                                <span class="n">cat</span> <span class="o">=</span> <span class="s1">&#39;Integer&#39;</span><span class="p">)</span>

<span class="n">breakpointValues</span> <span class="o">=</span> <span class="n">createBreakPoints</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
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<p>Now, we define the objective function as the approximate function value, and then add the constraints  as they were define earier in our mathematical formulation.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">##-add the objective function to the prob variable</span>
<span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span>

<span class="c1">##CONSTRAINTS##</span>
<span class="c1">#---------------#</span>
<span class="c1">#-define the approx function value as the sum of the inner product of </span>
<span class="c1">#-the breakpoint variables and the function value of where the breakpoints are</span>
<span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">optFunction</span><span class="p">(</span><span class="n">breakPointValues</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>

<span class="c1">#-define x as the sum of the inner product of the breakpoint variables and the corresponding values</span>
<span class="n">prob</span> <span class="o">+=</span> <span class="n">x</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>

<span class="c1">##Adjacency constraints</span>
<span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>

<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">numSegments</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
    <span class="c1">##print(i, &quot; + &quot;, i+1, &quot; &gt;= &quot;, i)</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>

<span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">numSegments</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="n">numSegments</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>

<span class="c1">##convex combination constraints</span>
<span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span>

<span class="c1">##only 1 segment selected</span>
<span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">segmentVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span>
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<p>We can also throw in another linear constraint just to show how we an use the discretization along with other constraints.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1">#y &lt;= 10-x</span>
<span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span> <span class="o">&lt;=</span> <span class="o">-</span><span class="mi">7</span><span class="o">/</span><span class="mi">10</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span>
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<p>Note that this constraint is defined with the <i>approximate</i> function value, so it's entirely possible that the optimal solution could violate the nonlinear counterpart to this constraint.</p>
<p>Below, we pull together the entire optimization program, and write it as a function.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">functionLinearApproximation</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="n">loggingFlag</span><span class="p">):</span> 
    <span class="c1">#-create the problem variable</span>
    <span class="n">prob</span> <span class="o">=</span> <span class="n">LpProblem</span><span class="p">(</span><span class="s1">&#39;Linearization Model&#39;</span><span class="p">,</span> <span class="n">LpMaximize</span><span class="p">)</span>

    <span class="c1">##primary decision variable##</span>
    <span class="c1">#---------------------------#</span>
    <span class="c1">#-domain variable</span>
    <span class="n">x</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">LpContinuous</span><span class="p">)</span>

    <span class="c1">#-variable for approximating nonlinear function</span>
    <span class="n">approxFunctionValue</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="p">(</span><span class="s2">&quot;w&quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">LpContinuous</span><span class="p">)</span>
    
    <span class="c1">#-create breakpoint variables</span>
    <span class="n">breakpointVars</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="o">.</span><span class="n">dicts</span><span class="p">(</span><span class="s2">&quot;partition var&quot;</span><span class="p">,</span> 
                                    <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)),</span>
                                    <span class="n">lowBound</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span>
                                    <span class="n">cat</span> <span class="o">=</span> <span class="s1">&#39;Continuous&#39;</span><span class="p">)</span>

    <span class="c1">#-create segment variables</span>
    <span class="n">segmentVars</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="o">.</span><span class="n">dicts</span><span class="p">(</span><span class="s2">&quot;segment var&quot;</span><span class="p">,</span>
                                    <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">)),</span>
                                    <span class="n">lowBound</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span>
                                    <span class="n">cat</span> <span class="o">=</span> <span class="s1">&#39;Integer&#39;</span><span class="p">)</span>
    
    <span class="c1">#-create a list of the breakpoint domain values</span>
    <span class="n">breakpointValues</span> <span class="o">=</span> <span class="n">createBreakPoints</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
    
    <span class="c1">##-add the objective function to the prob variable</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span>

    <span class="c1">##CONSTRAINTS##</span>
    <span class="c1">#-------------#</span>
    <span class="c1">#-define the approx function value as the sum of the inner product of </span>
    <span class="c1">#-the breakpoint variables and the function value of where the breakpoints are</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">optFunction</span><span class="p">(</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>

    <span class="c1">#-define x as the sum of the inner product of the breakpoint variables and the corresponding values</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">x</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>

    <span class="c1">#-Adjacency constraints</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>

    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">numSegments</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
        <span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">+</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>

    <span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">numSegments</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="n">numSegments</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>

    <span class="c1">#-convex combination constraints</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span>

    <span class="c1">#-only 1 segment selected</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">segmentVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span>

    <span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span> <span class="o">&lt;=</span> <span class="o">-</span><span class="mi">7</span><span class="o">/</span><span class="mi">10</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span>
    
    <span class="c1">#-solve the problem with default sovler</span>
    <span class="n">prob</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
    
    <span class="k">if</span> <span class="n">loggingFlag</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
        <span class="c1">#-if our logging flag is on, we write out all variable optimal values</span>
        <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Status:&quot;</span><span class="p">,</span> <span class="n">LpStatus</span><span class="p">[</span><span class="n">prob</span><span class="o">.</span><span class="n">status</span><span class="p">])</span>
        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">prob</span><span class="o">.</span><span class="n">variables</span><span class="p">():</span>
            <span class="nb">print</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="s2">&quot; = &quot;</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">varValue</span><span class="p">)</span>
            
    <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">varValue</span><span class="p">,</span> <span class="n">approxFunctionValue</span><span class="o">.</span><span class="n">varValue</span><span class="p">,</span> <span class="n">breakpointValues</span><span class="p">)</span>
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<p>Just below, we'll look at the output of the optimization function for 12 segments. Note that the breakpoints with positive values straddle the segment between them</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">functionLinearApproximation</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
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<pre>Status: Optimal
partition_var_0  =  0.0
partition_var_1  =  0.0
partition_var_10  =  0.0
partition_var_11  =  0.0
partition_var_12  =  0.0
partition_var_2  =  0.0
partition_var_3  =  0.0
partition_var_4  =  0.0
partition_var_5  =  0.0
partition_var_6  =  0.37889832
partition_var_7  =  0.62110168
partition_var_8  =  0.0
partition_var_9  =  0.0
segment_var_0  =  0.0
segment_var_1  =  0.0
segment_var_10  =  0.0
segment_var_11  =  0.0
segment_var_2  =  0.0
segment_var_3  =  0.0
segment_var_4  =  0.0
segment_var_5  =  0.0
segment_var_6  =  1.0
segment_var_7  =  0.0
segment_var_8  =  0.0
segment_var_9  =  0.0
w  =  3.1376907
x  =  5.5175847
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<pre>(5.5175847, 3.1376907)</pre>
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<h2 id="Results-and-Visualizations">Results and Visualizations<a class="anchor-link" href="#Results-and-Visualizations">&#182;</a></h2>
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<p>Below, we define a quick function to help us graph the results of the optimization for a given number of segments.</p>

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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">graphOptimization</span><span class="p">(</span><span class="n">numSegments</span><span class="p">):</span> 
    <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">breakpointValues</span> <span class="o">=</span> <span class="n">functionLinearApproximation</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">10</span><span class="p">],[</span><span class="mi">7</span><span class="p">,</span><span class="mi">0</span><span class="p">])</span>
    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">):</span>
        <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="p">:</span><span class="n">i</span><span class="o">+</span><span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="p">]),</span><span class="n">optFunction</span><span class="p">(</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])],</span> <span class="s1">&#39;ro-&#39;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">([</span><span class="n">x</span><span class="p">],</span> <span class="p">[</span><span class="n">y</span><span class="p">],</span> <span class="s1">&#39;go-&#39;</span><span class="p">)</span>
    <span class="n">sinFunction</span><span class="p">()</span>
    
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<p>Let's take a look at the the results from the optimization given 12 segmnets.</p>

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<h3 id="12-Segments">12 Segments<a class="anchor-link" href="#12-Segments">&#182;</a></h3>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">graphOptimization</span><span class="p">(</span><span class="mi">12</span><span class="p">)</span>
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<p>It's important to note that the given the small number of segments, we see a large amount of error in the approximation. You can see that in this case, the actual function value of the sin function would violate the constraint that the point be at or below the blue line. However, to remedy this, we can raise the number of segments, and hopefully get closer to the actual function value.</p>

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<h3 id="25-Segments">25 Segments<a class="anchor-link" href="#25-Segments">&#182;</a></h3>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">graphOptimization</span><span class="p">(</span><span class="mi">25</span><span class="p">)</span>
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<h3 id="100-Segments">100 Segments<a class="anchor-link" href="#100-Segments">&#182;</a></h3>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">graphOptimization</span><span class="p">(</span><span class="mi">100</span><span class="p">)</span>
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<h2 id="A-Better-Formulation">A Better Formulation<a class="anchor-link" href="#A-Better-Formulation">&#182;</a></h2>
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<p>While the model described above is certainly functional, there is another model that has more desirable properties for optimization. This model is described as model III in source [1], and we will work on the implementation here.</p>
<p>
First, we will take constraints (1)-(4), (5), (7), and (8) from the formulation above. We then add constraints that we will refer to as cumulative adjacency constraints. 
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$$ \sum^{k-1}_{j = i+1} z_j \leq \sum^{k}_{j=i+1} \xi_j \leq \sum^{k - 1}_{j = i} z_j \tag{10} \;\;\;\; \forall \ 1 \leq i \leq k-2$$
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<p>This constraint works similar to the adjacency constraints from the previous model, except that it works cumulatively. Let's take an example in which $i = 5$, and $k = 10$.</p>

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$$ z_6 + z_7 + z_8 + z_9 \leq \xi_6 + \xi_7 + \xi_8 + \xi_9 + \xi_{10} \leq z_5 + z_6 + z_7 + z_8 + z_9  $$
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<p>It ensures by groupwise constraints that any positive breakpoints straddle a selected segment.</p>
<p>
What makes this formulation better than the previous is because it is <i>locally ideal</i>. This means that the LP relaxation of the MIP has all extreme points of $z_i$ in {0,1}. For a small problem like ours, this improvement would probably not make a noticeable difference; however, it could be important for larger problems where speed is important, since it could dramatically reduce the size of the branch-and-bound tree.
</p><p>Regardless of speed improvement, we implement the alternate implementation below. Note that nearly everything stays the same, except for a few lines of adjacency constraints. 
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">cumulativeAdjacencyLinearApproximation</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="n">loggingFlag</span><span class="p">):</span> 
    <span class="c1">##create the problem variable</span>
    <span class="n">prob</span> <span class="o">=</span> <span class="n">LpProblem</span><span class="p">(</span><span class="s1">&#39;Linearization Model&#39;</span><span class="p">,</span> <span class="n">LpMaximize</span><span class="p">)</span>

    <span class="c1">##primary decision variable##</span>
    <span class="c1">#---------------------------#</span>
    <span class="c1">#-domain variable</span>
    <span class="n">x</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">LpContinuous</span><span class="p">)</span>

    <span class="c1">#-variable for approximating nonlinear function</span>
    <span class="n">approxFunctionValue</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="p">(</span><span class="s2">&quot;w&quot;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="kc">None</span><span class="p">,</span> <span class="n">LpContinuous</span><span class="p">)</span>
    
    <span class="c1">#-create breakpoint variables</span>
    <span class="n">breakpointVars</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="o">.</span><span class="n">dicts</span><span class="p">(</span><span class="s2">&quot;partition var&quot;</span><span class="p">,</span> 
                                    <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)),</span>
                                    <span class="n">lowBound</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span>
                                    <span class="n">cat</span> <span class="o">=</span> <span class="s1">&#39;Continuous&#39;</span><span class="p">)</span>

    <span class="c1">#-create segment variables</span>
    <span class="n">segmentVars</span> <span class="o">=</span> <span class="n">LpVariable</span><span class="o">.</span><span class="n">dicts</span><span class="p">(</span><span class="s2">&quot;segment var&quot;</span><span class="p">,</span>
                                    <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">)),</span>
                                    <span class="n">lowBound</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span>
                                    <span class="n">cat</span> <span class="o">=</span> <span class="s1">&#39;Integer&#39;</span><span class="p">)</span>
    
    <span class="c1">#-create a list of the breakpoint domain values</span>
    <span class="n">breakpointValues</span> <span class="o">=</span> <span class="n">createBreakPoints</span><span class="p">(</span><span class="n">numSegments</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
    
    <span class="c1">#-add the objective function to the prob variable</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span>

    <span class="c1">##CONSTRAINTS##</span>
    <span class="c1">#---------------#</span>
    <span class="c1">#-define the approx function value as the sum of the inner product of </span>
    <span class="c1">#-the breakpoint variables and the function value of where the breakpoints are</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">*</span> <span class="n">optFunction</span><span class="p">(</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="p">])</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>

    <span class="c1">#-define x as the sum of the inner product of the breakpoint variables and the corresponding values</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">x</span> <span class="o">==</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">*</span><span class="n">breakpointValues</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">))</span>

    <span class="c1">#-Adjacency constraints</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>

    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">numSegments</span> <span class="o">-</span> <span class="mi">1</span><span class="p">):</span>
        <span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">segmentVars</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">numSegments</span><span class="p">))</span> \
                <span class="o">&gt;=</span> <span class="nb">sum</span><span class="p">(</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span>
        <span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">j</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">numSegments</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="p">)</span> \
                <span class="o">&gt;=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">segmentVars</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span> <span class="o">+</span> <span class="mi">1</span><span class="p">,</span> <span class="n">numSegments</span><span class="p">)</span> <span class="p">)</span>

    <span class="n">prob</span> <span class="o">+=</span> <span class="n">breakpointVars</span><span class="p">[</span><span class="n">numSegments</span><span class="p">]</span> <span class="o">&gt;=</span> <span class="n">segmentVars</span><span class="p">[</span><span class="n">numSegments</span> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span>

    <span class="c1">#-convex combination constraints</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">breakpointVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span>

    <span class="c1">#-only 1 segment selected</span>
    <span class="n">prob</span> <span class="o">+=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">segmentVars</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">numSegments</span><span class="p">))</span> <span class="o">==</span> <span class="mi">1</span>

    <span class="n">prob</span> <span class="o">+=</span> <span class="n">approxFunctionValue</span> <span class="o">&lt;=</span> <span class="o">-</span><span class="mi">7</span><span class="o">/</span><span class="mi">10</span><span class="o">*</span><span class="n">x</span> <span class="o">+</span> <span class="mi">7</span>
    
    <span class="c1">#-solve the problem with default sovler</span>
    <span class="n">prob</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
    
    <span class="k">if</span> <span class="n">loggingFlag</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
        <span class="c1">#-if logging flag is on then write optimal values for variables</span>
        <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;Status:&quot;</span><span class="p">,</span> <span class="n">LpStatus</span><span class="p">[</span><span class="n">prob</span><span class="o">.</span><span class="n">status</span><span class="p">])</span>
        <span class="k">for</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">prob</span><span class="o">.</span><span class="n">variables</span><span class="p">():</span>
            <span class="nb">print</span><span class="p">(</span><span class="n">v</span><span class="o">.</span><span class="n">name</span><span class="p">,</span> <span class="s2">&quot; = &quot;</span><span class="p">,</span> <span class="n">v</span><span class="o">.</span><span class="n">varValue</span><span class="p">)</span>
            
    <span class="k">return</span> <span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">varValue</span><span class="p">,</span> <span class="n">approxFunctionValue</span><span class="o">.</span><span class="n">varValue</span><span class="p">,</span> <span class="n">breakpointValues</span><span class="p">)</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">cumulativeAdjacencyLinearApproximation</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
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<pre>Status: Optimal
partition_var_0  =  0.0
partition_var_1  =  0.0
partition_var_10  =  0.0
partition_var_11  =  0.0
partition_var_12  =  0.0
partition_var_2  =  0.0
partition_var_3  =  0.0
partition_var_4  =  0.0
partition_var_5  =  0.0
partition_var_6  =  0.37889832
partition_var_7  =  0.62110168
partition_var_8  =  0.0
partition_var_9  =  0.0
segment_var_0  =  0.0
segment_var_1  =  0.0
segment_var_10  =  0.0
segment_var_11  =  0.0
segment_var_2  =  0.0
segment_var_3  =  0.0
segment_var_4  =  0.0
segment_var_5  =  0.0
segment_var_6  =  1.0
segment_var_7  =  0.0
segment_var_8  =  0.0
segment_var_9  =  0.0
w  =  3.1376907
x  =  5.5175847
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<pre>(5.5175847,
 3.1376907,
 [0.0,
  0.8333333333333334,
  1.6666666666666667,
  2.5,
  3.3333333333333335,
  4.166666666666667,
  5.0,
  5.833333333333333,
  6.666666666666667,
  7.5,
  8.333333333333334,
  9.166666666666666,
  10.0])</pre>
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<p>For the next part in this series of nonlinear approximations, we'll cover how to implement a two dimensional grid, and then we'll cover how to create a binary indexed grid, which allows us to have a fine grid resolution at a low variable cost.</p>

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<h2 id="Sources">Sources<a class="anchor-link" href="#Sources">&#182;</a></h2>
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<p>[1] Padberg, M (2000) <a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1290963">Approximating Separable Nonlinear Functions Via Mixed Zero-One Programs</a></p>

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