The performance of a technical device is described by m quality measures fj: R R, j...

  

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The performance of a technical device is described by m quality measures fj: R" R, j = 1, ..., m. The smaller their values the better the performance is. The requirements of technical feasibility have been formulated in the form of s inequalities: 8i (x) 0, i = 1,..., s. The vector x R" represents design decisions and gi: R" R, i = 1, ..., All functions fj (.) and g; (-) are continuously differentiable. The builder is considering three different approaches to the design problem. Approach 1: She selects one of the performance measures, for example fi (x), to minimize, while keeping all the other measures fj (x) below some specified target levels bj, j = 2, ..., m. Approach 2: The builder creates an aggregate objective function m f(x) = wj fj (x), j=1 where w; 0 are some selected weights, at least one of which is positive. Approach 3: She uses target values b; and weights w; to create an aggregate objective function F(x)= max wj (fj(x) - bj) Ijsm representing the worst weighted excess over the targets. (a) For each of these three approaches formulate the resulting nonlinear opti- mization problem. (b) Assuming that constraint qualification is satisfied, formulate the first order necessary conditions of optimality for the corresponding problem. (c) Prove that for every solution obtained by Approach 2 we can find target levels bj such that the same solution is optimal in Approach 1 and in Approach 3. (d) Prove that if all functions are convex and a constraint qualification condition is satisfied, for every solution obtained by Approach 1 we can find weights w; such that the same solution is optimal in Approach 2 and Approach 3. (e) Prove that for every solution obtained by Approach 3 we can find a per- formance measure to minimize so that the same solution is optimal in Ap- proach 1. Prove that if the functions are convex then this optimal solution is also optimal in Approach 2. The performance of a technical device is described by m quality measures fj: R" R, j = 1, ..., m. The smaller their values the better the performance is. The requirements of technical feasibility have been formulated in the form of s inequalities: 8i (x) 0, i = 1,..., s. The vector x R" represents design decisions and gi: R" R, i = 1, ..., All functions fj (.) and g; (-) are continuously differentiable. The builder is considering three different approaches to the design problem. Approach 1: She selects one of the performance measures, for example fi (x), to minimize, while keeping all the other measures fj (x) below some specified target levels bj, j = 2, ..., m. Approach 2: The builder creates an aggregate objective function m f(x) = wj fj (x), j=1 where w; 0 are some selected weights, at least one of which is positive. Approach 3: She uses target values b; and weights w; to create an aggregate objective function F(x)= max wj (fj(x) - bj) Ijsm representing the worst weighted excess over the targets. (a) For each of these three approaches formulate the resulting nonlinear opti- mization problem. (b) Assuming that constraint qualification is satisfied, formulate the first order necessary conditions of optimality for the corresponding problem. (c) Prove that for every solution obtained by Approach 2 we can find target levels bj such that the same solution is optimal in Approach 1 and in Approach 3. (d) Prove that if all functions are convex and a constraint qualification condition is satisfied, for every solution obtained by Approach 1 we can find weights w; such that the same solution is optimal in Approach 2 and Approach 3. (e) Prove that for every solution obtained by Approach 3 we can find a per- formance measure to minimize so that the same solution is optimal in Ap- proach 1. Prove that if the functions are convex then this optimal solution is also optimal in Approach 2.