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algorithm analysis applied approximation buckling Calculus of Variations Chapter choose class of problems column components compression members computation conjugate directions Conjugate Gradient Method considered constraint functions constraint set convergence convex cost function cross section curve defined deflection denoted derivatives design parameter design variable determined direction of steepest displacement eigenvalue eigenvector element engineering Example Figure finite frame given inequality constraints interior Iteration Number lems linear programming linear programming problem linearly independent load matrix maximum minimize minimum point minimum weight natural frequency necessary conditions Newton Method nonlinear programming notation obtained optimal design problem optimization problems optimum paragraph pivot step Portal Frame prob relative minimum return to Step shown in Fig solution of NLP solved statically determinate steepest descent steepest-descent straints stress constraints struc structural design problem Table technique Theorem 6-7 theory tion tower treated truss vector violated weapon yields zero