Chapter 7

Nonlinear Optimization Models

Introduction

• The objective and/or the constraints are nonlinear functions of the decision variables.

• Select GRG Nonlinear in Solver• The Solver solution may not be optimal

Reasons for nonlinearity

• the effect of some input on some output is nonlinear• In pricing models where price is the decision variable,

and quantity sold is related to price, revenue is really price multiplied by a function of price – which is nonlinear

• Goodness of the fit requires minimizing sum of squared differences. The squaring introduces nonlinearity.

• Financial models try to achieve high return and low risk. Variance (or standard deviation) is used to meaure risk and it is nonlinear.

Local and global optimum

• For the figure graphed below, points A and C are called local maxima

• Only point A is the global maximum.

• If Solver finds point C first it will stop and present it as the best solution

Convex functions• A function of one variable is convex in a region if its slope (rate of

change) in that region is always nondecreasing.• i.e.. if a line drawn connecting two points the curve is always

below the curve.

Concave functions• A function is concave if its slope is always nonincreasing• In other words, if a line is drawn connecting two points the curve is

always above the line

Properties of concave and convex functions

• Sum of convex/concave functions is convex/ concave.

• Convex/concave functions multiplied by a positive constant is convex/concave

• Convex/concave functions multiplied by a negative constant will result in concave/convex.

Solver: GRG nonlinear

• Solver is guaranteed to find the global optimum if:– for maximization problems: (a) the objective

function is concave or the logarithm of the objective function is concave, and (b) the constraints are linear.

– Conditions for minimization problems: (a) the objective function is convex, and (b) the constraints are linear.

Solver: GRG nonlinear

• If the previous conditions are not true do the following:1. Try several possible starting (initial) values for the

changing cells, 2. Run Solver from each of these, and 3. Take the best solution Solver finds.

• In Solver this can be done using Multi-start option

Solver multi-start option