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algorithm approximating problem artiﬁcial variables assume axiaxi basic feasible solution Basis cg calculate Chapter coefﬁcients computational concave function convex combination convex function convex set cost deﬁne deﬁnition determine difﬁculty dual function dynamic programming enters the basis equations Euclidean space extreme points ﬁnd ﬁnding ﬁnite ﬁrst ﬁxed following problem func function h geometric programming global maximum gradient method gramming problem Hence hypersphere inequality initial integer iteration Lagrange multiplier Lagrangian function Leon Cooper line segment linear programming problem mathematical matrix Max z f(x maximize modiﬁed Nelder and Mead nonlinear programming problems nonnegative number of variables objective function obtain opti optimal solution optimal value optimization problem original problem posynomial problem given proﬁt quadratic programming satisﬁed separable programming set of points shown in Fig simplex method single variable Solving eqs step straint Suppose surplus variables technique tion unconstrained Vf(x x'Px