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Unconstrained and Linearly Constrained Extrema
Existence of extreme points
Unconstrained local minima and maxima
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accordingly affords a strict Chapter closure condition cone G Consequently constant converges convex cone convex function convex set corresponding critical point defined denoted derived cone described in Theorem determined differentiable at x0 eigenvalue eigenvector equations Euclidean space example finite dimensional follows formula function F function f(x,y ga(x given in Exercise given in Theorem gradient Hence holds implicit function theorem Lagrange multiplier rule Lemma linear subspace linearly independent minimizes f minimizes the function minimum to f Moreover neighborhood normal point null space Observe obtain optimization orthogonal orthonormal outer normal point of F positive numbers problem of minimizing pseudoinverse quadratic form Rayleigh quotient regular point relative results given Section sequence xq set of multipliers set of points Show solution strict minimum Suppose symmetric matrix tangent cone tangent space tangent vector tangential constraints Theorem 3.1 theory unit vector vector h x*Ax x0 affords x0 minimizes xy-plane zero