# Constrained optimization

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## Outline

• The following topics will be covered in this lecture:
• Linear programming problems
• Nonlinear programming problems

## Motivation

• In our previous discussion of optimization, we focused on optimization without constraints.

• This arises commonly in statistical estimation problems and is actually a simpler case of the constrained optimization problem.
• Constrained optimization also commonly arises in statistical estimation.

• For instance, suppose that we need to estimate the variance of some distribution by an optimization routine.
• A fully unconstrained optimization could (in principle) lead to nonsensical values for the variance.
• The variance is defined to be strictly non-negative, where a solution giving $$\sigma^2<0$$ would cause obvious errors in our analysis.
• In this final discussion of optimization, we will consider how constraints are introduced to our optimization framework.

• This will lead us into two classes of constrained optimization problems:

• Linear Programming (LP) problems; and
• Nonlinear Programming (NLP) problems.
• After introducing some general concepts, we will consider several techniques that can be used in the R language to handle such problems.

## Linear Programming problems

• A LP optimization is a method to find the solution to an optimization problem with:
1. a linear objective function, and
2. constraints in the form of linear equalities and linear inequalities.
• When we say a linear objective function, we are referring to a linear $$f$$ such that \begin{align} f:\mathbb{R}^{n} &\rightarrow \mathbb{R}\\ \pmb{x}&\rightarrow \pmb{a}^\top \pmb{x} \end{align} as this must be represented by a linear map (matrix multiplication) that transfers $$\pmb{x}\in\mathbb{R}^n$$ to a real value.
• This is thus given precisely by a vector product as above for some $$\pmb{a}\in \mathbb{R}^n$$ as we can treat this product as \begin{align} \underbrace{\pmb{a}^\top}_{1\times n} \underbrace{\pmb{x}}_{n \times 1} = \underbrace{y}_{1\times 1}\in \mathbb{R} \end{align}
• Respectively, when we say that we have linear constraints, these can describe, e.g., \begin{align} x_1 + x_2 \leq 1 & & x_1 - x_2 \leq 1 & & -x_1 + x_2 \leq 1 & & -x_1 - x_2 \leq 1. \end{align}
• Generally, therefore, a LP optimization has a region of acceptable solutions defined by a convex polyhedron, which is a set made by the intersection of finitely many half-spaces. Courtesy of: J. Nocedal and S. Wright. Numerical optimization. Springer Science & Business Media, 2006.

• This convex polyhedron is denoted $$\Omega$$ and called the feasible region of the problem.
• The objective of linear programming is to find a point in the feasible region where the objective function reaches a minimum or maximum value.

### Linear Programming problems

• A representative LP problem can be expressed as finding $$\pmb{x}^\ast$$ such that

\begin{align} f(\pmb{x}^\ast) &= \max_{\pmb{x}\in \Omega} \pmb{a}^\top \pmb{x},\quad \text{subject to:}\\ \\ &\mathbf{C}\pmb{x} \leq \pmb{b},\\ &\pmb{x} \geq \pmb{0}, \end{align} where $$\pmb{b}$$ is a vector of known coefficients and $$\mathbf{C}$$ is a known matrix of the coefficients in the constraints.

• Because the objective function is linear, it is both convex and concave simultaneously.

• Therefore, as long as the constraints are consistent,

• and provide a bounded feasible region as seen before,
• we can obtain a global minimum or maximum for such a problem on the feasible region boundary.

• Although the linear objective function is a strong constraint on the problem

• this type of problem can represent a variety of practical optimization scenarios…

### Linear Programming problems

• For example, suppose that a farmer has a piece of farm land, say $$L$$ $$\mathrm{km}^2$$, to be planted with either wheat or barley or some combination of the two.

• The farmer has a limited amount of fertilizer, $$F$$ kg, and pesticide, $$P$$ kg.

• Every square kilometer of wheat requires $$F_1$$ kilograms of fertilizer and $$P_1$$ kilograms of pesticide.

• On the other hand, every square kilometer of barley requires $$F_2$$ kilograms of fertilizer and $$P_2$$ kilograms of pesticide.

• Let $$S_1$$ be the selling price of wheat per square kilometer, and $$S_2$$ be the selling price of barley.

• If we denote the area of land planted with wheat and barley by $$x_1$$ and $$x_2$$ respectively, then profit can be maximized by choosing optimal values for $$x_1$$ and $$x_2$$.

### Linear Programming problems

• The example problem can be expressed with the following linear programming problem in the standard form:

\begin{align} \text{Maximize: } S_{1}\cdot x_{1}+S_{2}\cdot x_{2} & &\text{(maximize the revenue)}\\ \text{Subject to:}\\ x_{1}+x_{2}\leq L & & \text{(limit on total area)}\\ F_{1}\cdot x_{1}+F_{2}\cdot x_{2}\leq F & & \text{(limit on fertilizer)}\\ P_{1}\cdot x_{1}+P_{2}\cdot x_{2}\leq P & & \text{(limit on pesticide)}\\ x_{1}\geq 0,x_{2}\geq 0 & & \text{(cannot plant a negative area).} \end{align}

• Alternatively, in matrix form we have this written equivalently as

\begin{align} \text{Maximize: }\pmb{S}^\top \pmb{x} \\ \text{Subject to:}\\ \begin{pmatrix} 1 & 1\\ F_1 & F_2 \\ P_1 & P_2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} \leq \begin{pmatrix}L \\ F \\ P \end{pmatrix} \\ \begin{pmatrix}x_1 \\ x_2 \end{pmatrix} \geq \begin{pmatrix} 0 \\ 0 \end{pmatrix} \end{align}

### Linear Programming problems

• Linear programming problems as above can be solved in R by using the R wrapper for the GNU Linear Programming Kit (GLPK).

• This comes in the package Rglpk below:

require(Rglpk)

• As an example, we will show how to solve the following problem,

\begin{align} \text{Maximize: }\begin{pmatrix}2 \\ 4\end{pmatrix}^\top \begin{pmatrix}x_1 \\ x_2\end{pmatrix}\\ \text{Subject to:} & & \begin{pmatrix} 3 \\ 4\end{pmatrix}^\top \begin{pmatrix}x_1 \\ x_2\end{pmatrix} \leq 60 & & \begin{pmatrix}x_1 \\ x_2\end{pmatrix} \geq \pmb{0} \end{align}

• We will use the Rglpk_solve_LP with the following arguments:

• obj - a numeric vector representing the objective coefficients.
• mat - a matrix of constraint coefficients.
• dir - a character vector with the directions of the constraints, "<", "<=", ">", ">=", or "==".
• rhs - a numeric vector representing the right hand side of the constraints.

### Linear Programming problems

Rglpk_solve_LP(obj = c(2, 4),
mat = matrix(c(3, 4), nrow = 1),
dir ="<=",
rhs = 60,
max = TRUE)

$optimum  60$solution
  0 15

$status  0$solution_dual
 -1  0

$auxiliary$auxiliary$primal  60$auxiliary$dual  1$sensitivity_report
 NA


### Linear Programming problems

• The geometry of the LP problem can be understood where the output of the objective function $$f$$ is given as a hyper-plane above the $$x_1,x_2$$ plane.
• The constraints likewise define the convex polyhedron through the intersection of the corresponding half-hyper-planes.
• Correspondingly, the maximum is attained where the polyhedron intersects the hyper-plane of the objective function.
• This is visualized to the right for this problem.