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admissible direction admissible linking admissible transfer element ajvj apply attack backward stability calculate chain chain rule concave constant constraint continuous function continuously differentiable convex convex set corner solutions corresponding Danskin decreasing Dedekind cut denote difference quotients directional derivative elementary eligible row equation Euclidean space example Exercise follows formula formula 17 Gibbs lemma given Harlan Mills Hence inequality ith row keZi Lagrange multiplier principle Lagrange multiplier theorem left side lemma is proved linking structure Max MinF(x Max-Min problem Max-Min-Max problem minimum optimal partial derivatives percentage vulnerable player Proof quantity reaches unity result right side satisfying Section segment side conditions smooth function solution matrix solved strictly increasing Suppose target Theory of Max-Min trivial unique vector weapons of type weapons systems y e Y(x y-player Y(xm yields the maximum