Linear and Convex Optimization. Michael H. Veatch

rel="nofollow" href="#fb3_img_img_9da282e6-f91e-5ca0-8fb2-235bb38acdd5.png" alt="images"/> if false (decide not to take the action). A variable whose possible values are is called a binary variable. An integer program with all binary variables is called a binary program.

      General Form of an Integer Program

      1.4.3 Nonlinear Programs

An optimization problem where the objective function and constraints may be nonlinear is called a nonlinear program. The variables are assumed to be continuous, as in a linear program. Thus, nonlinear programs are more general than linear programs, but do not include integer programs. While linear programs can be described in matrix notation, nonlinear programs are described in terms of functions.

      General Form of a Nonlinear Program

      We have not stated the nonnegativity constraints separately. However, a nonnegativity constraint can be included in the functional constraints if needed.

      The ability to solve a nonlinear program depends on the type of objective function and constraints . A class of more easily solved nonlinear programs called convex programs is discussed in Chapters 14 and 15. Also, Chapter 3 shows how certain piece‐wise linear convex programs can be converted into linear programs, which is useful because convex programs are somewhat harder to solve than linear programs.

      Problems

      For Exercises 1–6, solve the linear program graphically.

      For Exercises 6–8, solve the integer program graphically.

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      4 a) Solve as stated. b) Change “min” to “max” and solve.

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      9 For the linear program (1.1)Suppose the objective is to minimize the cost of aid given in 1.2. What is the optimal solution? Explain why minimizing cost is not a reasonable objective for this problem.Find an objective function for which , is optimal. Show graphically that this point is optimal.