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Chapter 7

Nonlinear Optimization Models. Chapter 7. 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.

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Chapter 7

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  1. Nonlinear Optimization Models Chapter 7

  2. 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

  3. 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.

  4. Local and global optimum • For the figure graphed below, points A and C are called local maxima • Only point Ais the global maximum. • If Solver finds point C first it will stop and present it as the best solution

  5. 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.

  6. 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

  7. 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.

  8. Solver: GRG nonlinear • Solver is guaranteed to find the global optimumif: • 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.

  9. Solver: GRG nonlinear • If the previous conditions are not true do the following: • Try several possible starting (initial) values for the changing cells, • Run Solver from each of these, and • Take the best solution Solver finds. • In Solver this can be done using Multi-start option

  10. Solver multi-start option

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