MIed Termination Criteria: Heuristics:| MIest Termination in iSIGHT幽甥 Simulated Annealing-cooling schedule: T(k=f(k, To) Be careful with iSIGHT: the last solution is not necessarily the best one Search stops when Need to look back over all the solutions in the monitor to T(k)≤,wheeε>0, find the"optimum but smal Feasibility paramete 3= infeasible search broadly search locally Tabu search termination 8= feasible and equal to the best design found so fal Usually after a predefined number of iterations 9=feasible and the best design found so far Best solution found is reported No guarantee of optimality The "optimum" will be the last solution with feasibility=9 Experimentation gives confidence e Massachusetts Institute of Technology- Prof de Weck and Prof Willcox e 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 Post-Optimality Analysis E07 Mlesd Lagrange Multipliers 507 The values of the Lagrange multipliers at the optimal Already talked about sensitivity analysis( Lecture 9) solution, i, give information on the constraints How does optimal solution change as a parameter is varied? If y is zero, then constraint j is inactive How does optimal solution change as a design If a is positive then constraint is active variable value is varied? The value of the /h Lagrange multiplier tells you by How does optimal solution change as constraints how much the objective function will change if the are varied? constraint is varied by a small amount Also would like to understand key drivers in optimal 0(x)=-4g(x) i is a vector containing the m LMs g is a vector containing all m constraints(inequality+equality) a Massachusetts Institute of Technology -Prof de Weck and Prof Willcox e Massachusetts Insttute of Technology. Prof de Weck and Prof Willcox Engineening Systems Division and Dept of Aeronautics and Astronautics Engineering Systems DiMsion and Dept of Aeronautics and Astronautics13 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Termination Criteria: Heuristics Termination Criteria: Heuristics • Simulated Annealing - cooling schedule: T(k)=f(k, To) To 0 Search stops when T(k)<ε , where ε>0, but small • Tabu search termination • Usually after a predefined number of iterations • Best solution found is reported • No guarantee of optimality • Experimentation gives confidence search broadly search locally 14 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Termination in Termination in iSIGHT • Be careful with iSIGHT: the last solution is not necessarily the best one • Need to look back over all the solutions in the monitor to find the “optimum” • Feasibility parameter: 3 = infeasible 7 = feasible 8 = feasible and equal to the best design found so far 9 = feasible and the best design found so far • The “optimum” will be the last solution with feasibility=9 15 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Post-Optimality Analysis Optimality Analysis • Already talked about sensitivity analysis (Lecture 9) – How does optimal solution change as a parameter is varied? – How does optimal solution change as a design variable value is varied? – How does optimal solution change as constraints are varied? • Also would like to understand key drivers in optimal design 16 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Lagrange Multipliers Lagrange Multipliers • The values of the Lagrange multipliers at the optimal solution, λj*, give information on the constraints. • If λj* is zero, then constraint j is inactive. • If λj* is positive, then constraint j is active. • The value of the jth Lagrange multiplier tells you by how much the objective function will change if the constraint is varied by a small amount: T ∂ =− ∂ J g ( *) ( *) x x λ – λ is a vector containing the m LMs – is a vector containing all g m constraints (inequality+equality)