1 Optimization

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NOTES OF LIGTHWEIGTH DESIGN OF MECHANICAL STRUCTURE Chiara Moreschini, AA 2022/23 MODULE 1: OPTIMIZATION • A design meeting all the requirements is called feasible • Active constraint = inequality constraint that satisfies the related equality (=) à once the optimum solution has been reached, the active constraint is the one constraining the problem Inactive constraint = inequality constraint that satisfies the related strict inequality (>,<) • Discrete variable = it has to be chosen from a given finite set of numbers (ex. Manufacturing sizes) à discrete non-linear programming problem. The simplest procedure to solve discrete problems is assuming the variables continuous and solve the continuous problem. Then the nearest discrete values are assigned to the variables and the design is checked for feasibility. With a few trials, the best feasible design closed to the continuum optimum is obtained. • Excel solver à direct method (search method) • Problems with equality constraints à Lagrange theorem • Problems with inequality constraints à KKT theorem: inequalities turned into equalities adding the slack variables, then apply same procedure of Lagrange theorem. Using the KKT, when deriving the Lagrange function we obtain always the switching condition (su=0 à s=0 or u=0) from which we obtain 2 sets of solutions s=0 à active constraint à must have u>=0 u=0 à inactive constraint à must have s>=0 So if s=0 the Lagrange multiplier must be non-negative; however this condition can become a non-positivity condition depending on the formulation of the problem (like in the Excel solver). When imposing s=0 or u=0, one of the following conditions (u>=0 or s>=0 respectively) could be not satisfied, so in that case the KKT theorem’s hypotheses are not satisfied and the correct set of…