160Fa04Midterm_Solutions - Department of Industrial...

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Unformatted text preview: Department of Industrial Engineering & Operations Research IEOR160 Operations Research I Exam 1 10/13/2004 Name: Grade: Closed book, closed notes exam. No cheatsheets. Programmable calculators not allowed. 1. (15 points) Determine whether the following statements are true or false . T F A concave function cannot have a minimizer over an equality constrained feasible region. False T F If the objective function of an optimization problem is convex and the feasible region is convex, then it is a minimization problem. False T F If f is a continuously differentiable function, then all of its local maxima are among its stationary points. True T F If x is a local maximum of a concave function, then there exists a direction vector d for which the directional derivative at x is negative. False, the directional derivative is zero T F For a KKT point, if the Lagrange multiplier of a constraint is zero, then the constraint is inactive at this point. False 2. (15 points) Definition: A function f : IR n IR is strictly quasiconvex on S IR n if for each x, y S such that x 6 = y the following inequality holds: f ( x + (1- ) y ) < max { f ( x ) , f ( y ) } for all < < 1 Prove that if f is strictly quasiconvex on S , then a local minimum of f on S is also a global minimum of f on S . Suppose x S such that there exists y S with f ( y ) < f ( x ) (i.e., x is not a global minimum). Let (0 , 1). Then, from definition, we have that f ( x +(1- ) y ) < max { f ( x ) , f ( y ) } = f ( x ). Hence, given any > 0, one can find a (0 , 1) such that z = (1- ) y + x lies in the -neighbourhood of x , and we know that f ( z ) < f ( x ). This implies that x cannot be a local minimum. Hence, any local minimum should be a global minimum too....
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160Fa04Midterm_Solutions - Department of Industrial...

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