Linear and Convex Optimization. Michael H. Veatch

Linear and Convex Optimization

integer values. The function being optimized is the objective function. The domain over which the function is optimized is the set of points that satisfy the constraints, which can be equations or inequalities. We distinguish two kinds of constraints. Functional constraints can have any form, while variable bounds restrict the allowable values of one variable. Many problems have nonnegativity constraints as their variable bounds. In (1.1), the food constraint images

is listed with the functional constraints, making three functional constraints plus two nonnegativity constraints (variable bounds), but it is also a bound, so the problem can also be said to have two functional constraints and three bounds.

Let images be the decision variables and images images satisfies the constraints images be the solution set of the constraints. Departing somewhat from the usual meaning of a “solution” in mathematics, we make the following definitions

A solution to a mathematical program is any value of the variables.

A feasible solution is a solution that satisfies all constraints, i.e. .

The feasible region is the set of feasible solutions, i.e. .

The value of a solution is its objective function value.

An optimal solution is a feasible solution that has a value at least as large (if maximizing) as any other feasible solution.

To solve a mathematical program images is to find an optimal solution images and the optimal value images , or determine that none exists. We can classify each mathematical program into one of three categories.

Existence of Optimal Solutions When solving a mathematical program:

The problem has one or more optimal solutions (we include in this category having solutions whose value is arbitrarily close to optimal, but this distinction is rarely needed), or

The problem is an unbounded problem, meaning that, for a maximization, there is a feasible solution with value for every , or

The problem is infeasible: there are no feasible solutions, i.e. .

An unbounded problem, then, is one whose objective function is unbounded on images (unbounded above for maximization and below for minimization). One can easily specify constraints for a decision problem that contradict each other and have no feasible solution, so infeasible problems are commonplace in applications. Any realistic problem could have bounds placed on the variables, leading to a bounded feasible region and a bounded problem. In practice, these bounds are often omitted, leaving the feasible region unbounded. A problem with an unbounded feasible region will still have an optimal solution for certain objective functions. Unbounded problems, on the other hand, indicate an error in the formulation. It should not be possible to make an infinite profit, for example.

Geometric representation of the problem which has optimal solution for dashed objective but is unbounded for dotted objective.

Figure 1.3 Problem has optimal solution for dashed objective but is unbounded for dotted objective.

Example 1.1 (Unbounded region) Consider the linear program

(1.3) equation

The feasible region is shown in Figure 1.3; the dashed line has an objective function value of 210. Sweeping this line down, the minimum value occurs at the corner point images with an optimal value of 180. The feasible region is unbounded. Now suppose the objective function is changed to images . The dotted line shows the contour line images images ; sweeping the line upward decreases the objective without bound, showing that the problem is unbounded.

1.4 Classes of Mathematical Programs

In Section 1.2 we introduced the concept of a linear program. This section introduces algebraic and matrix notation for linear programs. It then defines the three main classes of mathematical programs: linear programs, integer programs, and nonlinear programs. Each of these will be studied in later chapters, though for nonlinear programs we will focus on the more tractable subclass of convex programs. A mathematical program with variables images and objective function images can be stated abstractly in terms of a feasible region images as

However, we will always describe the feasible region using constraint equations or inequalities. The notation for the constraints is introduced in the following text.

1.4.1 Linear Programs

We have already seen two examples of linear programs.

General Form of a Linear Program

There are images decision variables, images and images functional constraints. The constraints can use a mixture of “ images ”, “ images ”, and “ images ”. Each variable may have the bound images , images , or no bound, which we call unrestricted in sign (u.r.s.). The distinguishing characteristics of a linear program are (i) the objective function and all constraints are linear functions and (ii) the variables are continuous, i.e. fractional values are allowed. They are often useful

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