MAE 552 – Heuristic Optimization Lecture 1 January 23, 2002

The optimization problem is then: Find values of the variables that minimize or maximize the objective function while satisfying the constraints. The standard form of the constrained optimization problem can be written as: Minimize: F(x)objective function Subject to:g j (x)  0j=1,m inequality constraints h k (x)  0k=1,l equality constraints x i lower  x i  x i upper i=1,nside constraints where x=(x 1, x 2, x 3, x 4, x 5,x n )design variables

Conditions for Optimality Unconstrained Problems 1.  F(x)=0The gradient of F(x) must vanish at the optimum 2.Hessian Matrix must be positive definite (i.e. all positive eigenvalues at optimum point).

Conditions for Optimality Unconstrained Problems A positive definite Hessian at the minimum ensures only that a local minimum has been found The minimum is the global minimum only if it can be shown that the Hessian is positive definite for all possible values of x. This would imply a convex design space. Very hard to prove in practice!!!!

Conditions for Optimality Constrained Problems Kuhn Tucker Conditions 1.x * is feasible 2. j g j = 0j=1,m 3. These conditions only guarantee that x * is a local optimum.

Conditions for Optimality Constrained Problems In addition to the Kuhn Tucker conditions two other conditions two other conditions must be satisfied to guarantee a global optima. 1.Hessian must be positive definite for all x. 2.Constraints must be convex. A constraint is convex if a line connecting any two points in the feasible space travels always lies in the feasible region of the design space.