Numerical Optimization Techniques for Engineering Design: With Applications

a₁ active constraints airfoil analysis approach calculated Chap coefficient computer storage concepts consider constraint boundary constraint functions constraint violations convergence dependent variables design problem design space design variables efficiency equality constraints Equation example extended penalty exterior penalty function F₁ feasible directions feasible region Figure function evaluations geometry given by Eq golden section method Hessian matrix hypersphere improve inequality constraints iteration Kuhn-Tucker conditions Lagrange multipliers linear programming load LP problem method of feasible Newton's method Nonlinear Programming number of design numerical optimization objective function one-dimensional search optimization process optimum design penalty function method penalty parameter polynomial interpolation pseudo-objective function push-off factors quadratic reduced gradient method search direction shown in Fig side constraints simplex algorithm simplex method slack variables solution solve steepest descent structural optimization subproblem Taylor series truss two-variable function space unconstrained minimization update vector X₁ X₂ zero