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Unformatted text preview: 1 Nonlinear Optimization Models k MGMT 306: Management Science Krannert Graduate School of Management Purdue University 2 Nonlinear Optimization Max or Min { Objective function} Subject to Constraint 1 Constraint 2 . . Nonnegativity constraints Possibly nonlinear 3 Examples of Nonlinear Programming Models Effect of advertising on sales : Beyond some advertising level, extra advertising dollars have much less effect on sales than that of initial advertising dollars. Pricing models : Revenue is price multiplied by quantity sold. Price is typically the decision variable, while quantity sold is related to price through a demand function. Portfolio optimization models : The risk of investment is typically measured as the variance (or standard deviation) of the portfolio, and it is a nonlinear function of the decision variables (the investment amounts). Location analysis models: When looking for the best site for a warehouse, cell phone tower, fire station, etc., distances are often measured using the Pythagorean theorem. 4 Nonlinear Optimization Solutions NLP Solver sometimes obtains a suboptimal solution. A local optimal solution is better than all nearby points. A global optimal solution is the best point in the entire feasible region. For some NLP problems, Solver can get stuck at a local optimal solution and never find the global optimum. A B C D Local maximum Global maximum Global minimum Local minimum 5 Local vs. Global Optimal Solutions Feasible Region X1 X2 optimal solution 6 Local vs. Global Optimal Solutions X1 X2 A Local optimal solution D F G Local and global optimal solution E B C A global optimal solution is also a local optimal solution 7 Convex Functions NLP Solver can solve certain types of NLPs globally. A function of one variable is convex in a region if its slope (rate of change) in that region is always nondecreasing. Equivalently, a function is convex if no line segment connecting two points on the curve goes below the function. 8 Concave Functions A function of one variable is concave in a region if its slope (rate of change) in that region is always nonincreasing. Equivalently, a function is concave if no line segment connecting two points on the curve is above the function. 9 Common Convex and Concave Functions Examples of convex functions: Examples of concave functions: , where 1, 0 and 0....
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 Spring '08
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 Management

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