Titlebook: Nondifferentiable Optimization; M. L. Balinski,Philip Wolfe Book 1975Latest edition Springer-Verlag Berlin Heidelberg 1975 978-0-444-11008

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Minimax solution of non-linear equations without calculating derivatives,s of the functions, and solution of the resulting linear systems in the minimax sense subject to bounds on the solutions. Approximations to the matrix of derivatives are updated by using the Broyden [2] rank-one updating formula. It is shown that the method has good convergence properties. Some numerical examples are given.

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Nondifferentiable optimization via approximation, the nondifferentiable function by a smooth function and is related to penalty and multiplier methods for constrained minimization. Some convergence results are given and the method is illustrated by means of examples from nonlinear programming.

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The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices,derived. Such a sum is convex but not necessarily everywhere differentiable. A convergent procedure is presented for determining a minimizing point of any such sum subject to the condition that the trace of the matrix is held constant. An implementation of this procedure is described and numerical r

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Using duality to solve discrete optimization problems: Theory and computational experience,e traveling salesman problem. In this paper a general class of discrete optimization problem is given for which dual problems of this type may be derived. We discuss the use of dual problems for obtaining strong bounds, feasible solutions, and for guiding the search in enumeration schemes for this c

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Minimax solution of non-linear equations without calculating derivatives,roblem have continuous first derivatives, but no expression for these derivatives is available. The method is based on successive linear approximations of the functions, and solution of the resulting linear systems in the minimax sense subject to bounds on the solutions. Approximations to the matrix

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