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adjoint equations adjoint variable algorithm apply base point boundary conditions Calculus of Variations computed consider constant constrained derivatives constraint equations contour cost Cramer's Rule decision policy decision variables derivative of f differential equation dynamic programming equa equal Euler-Lagrange equation example expression extremum feasible region feasible solutions Figure function evaluations geometric programming gradient direction Hamiltonian Hence independent variables inequality constraints initial integral Kuhn-Tucker conditions Lagrange multiplier linear programming locate maximize maximum principle method minimized mixing coefficients necessary conditions non-negative nonlinear nonlinear programming objective function obtained optimal policy optimum parameter optimization problem partial derivatives pattern search perturbation posynomial Practicing Engineers prob programming problem quadratic programming recycle region of feasible satisfy selected simplex algorithm solution procedure solved specified stage straint Substitution surplus variables system equations test point Theorem tion trajectory optimization problem vanish x1 and x2 xv x2 yields zero