MIest Kuhn-Tucker Conditions 16888 Interpretation Mlesd Optimization for Engineering 50. 3 Problems Condition 1: the optimal design satisfies the constraints Most engineering problems have a complicated design space, usually with several local optima Condition 2: if a constraint is not precisely satisfied, then Gradient-based methods can have trouble converging to the the corresponding Lagrange multiplier is zero orrect solution the h Lagrange multiplier represents the sensitivity of the Heuristic techniques offer absolutely no guarantee of objective function to the h constraint optimality, neither global nor local can be thought of as representing the tightness"of the Your post-optimality analysis should address the question constraint How confident are you that you have found the global if i, is large, then constraint is important for this solution optimum? Condition 3: the gradient of the lagrangian vanishes at Do you actually care? the optimum @ Massachusetts Institute of Technology -Prof de Weck and Prof Willcox G Massachusetts Insttute of Technology .Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics Engineering Systems Division and Dept of Aeronautics and Astronautics Mlesd Optimization for Engineering 650 3 MleSaTermination Criteria: Gradient-Based 50. 3 Problems Gradient-based algorithm is terminated when Usually cannot guarantee that absolute optimum is found an acceptable solution is found local optima numerical ill-conditioning a gradient-based techniques should be started from Need to decide. several initial solutions when an acceptable solution is found o best solution from a heuristic technique should be checked with kt conditions or used as an initial when to stop the algorithm with no acceptable condition for a gradient-based algorithm solution Can determine mathematically if have relative minimum but when progress is unreasonably slow Kuhn- Tucker conditions are only sufficient if the problem is when a specified amount of resources have been used(time number of iterations, etc. It is very important to interrogate the"optimum" solution when an acceptable solution does not exist a Massachusetts Institute of Technology - Prof de Weck and Prof Willcox when the iterative process is cycling @Massachusetts Insttute of Technology. Prof de Weck and Prof Willcox Engineening Systems Division and Dept of Aeronautics and Astronautics Aeronautics and astronautics5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Kuhn-Tucker Conditions: Tucker Conditions: Interpretation Interpretation Condition 1: the optimal design satisfies the constraints Condition 2: if a constraint is not precisely satisfied, then the corresponding Lagrange multiplier is zero – the jth Lagrange multiplier represents the sensitivity of the objective function to the jth constraint – can be thought of as representing the “tightness” of the constraint – if λj is large, then constraint j is important for this solution Condition 3: the gradient of the Lagrangian vanishes at the optimum 6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Optimization for Engineering Optimization for Engineering Problems Problems • Most engineering problems have a complicated design space, usually with several local optima • Gradient-based methods can have trouble converging to the correct solution • Heuristic techniques offer absolutely no guarantee of optimality, neither global nor local • Your post-optimality analysis should address the question: – How confident are you that you have found the global optimum? – Do you actually care? 7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Optimization for Engineering Optimization for Engineering Problems Problems • Usually cannot guarantee that absolute optimum is found – local optima – numerical ill-conditioning Æ gradient-based techniques should be started from several initial solutions Æ best solution from a heuristic technique should be checked with KT conditions or used as an initial condition for a gradient-based algorithm • Can determine mathematically if have relative minimum but Kuhn-Tucker conditions are only sufficient if the problem is convex • It is very important to interrogate the “optimum” solution 8 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Termination Criteria: Gradient Termination Criteria: Gradient-Based Gradient-based algorithm is terminated when ... an acceptable solution is found OR algorithm terminates unsuccessfully Need to decide: • when an acceptable solution is found • when to stop the algorithm with no acceptable solution – when progress is unreasonably slow – when a specified amount of resources have been used (time, number of iterations, etc.) – when an acceptable solution does not exist – when the iterative process is cycling