Nondifferentiable Optimization: Motivations and Applications Proceedings of an IIASA (International Institute for Applied Systems Analysis) Workshop on Nondifferentiable Optimization Held at Sopron, Hungary, September 17–22, 1984 Lecture Notes in Economics and Mathematical Systems Series, Vol. 255

The International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Austria, has been involved in research on nondifferentiable optimization since 1976. IIASA-based East-West cooperation in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimi­ zation has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition, and to review recent developments in this field, IIASA held a Workshop on Nondifferentiable Optimization in Sopron (Hungary) in September 1964. The aims of the Workshop were: 1. To discuss the state-of-the-art of nondifferentiable optimization (NDO), its origins and motivation; 2. To compare-various algorithms; 3. To evaluate existing mathematical approaches, their applications and potential; 4. To extend and deepen industrial and other applications of NDO. The following topics were considered in separate sessions: General motivation for research in NDO: nondifferentiability in applied problems, nondifferentiable mathematical models. Numerical methods for solving nondifferentiable optimization problems, numerical experiments, comparisons and software. Nondifferentiable analysis: various generalizations of the concept of subdifferen­ tials. Industrial and other applications. This volume contains selected papers presented at the Workshop. It is divided into four sections, based on the above topics: I. Concepts in Nonsmooth Analysis II. Multicriteria Optimization and Control Theory III. Algorithms and Optimization Methods IV. Stochastic Programming and Applications We would like to thank the International Institute for Applied Systems Analysis, particularly Prof. V. Kaftanov and Prof. A.B. Kurzhanski, for their support in organiz­ ing this meeting.

I. Concepts in Nonsmooth Analysis.- Attempts to Approximate a Set-Valued Mapping.- Miscellanies on Nonsmooth Analysis and Optimization.- Bundle Methods, Cutting-Plane Algorithms and ?-Newton Directions.- The Solution of a Nested Nonsmooth Optimization Problem.- Variations on the Theme of Nonsmooth Analysis: Another Subdifferential.- Lipschitzian Stability in Optimization: The Role of Nonsmooth Analysis.- Upper-Semicontinuously Directionally Differentiable Functions.- A New Approach to Clarke’s Gradients in Infinite Dimensions.- II. Multicriteria Optimization and Control Theory.- A Nondifferentiable Approach to Multicriteria Optimization.- Application of a Subdifferential of a Convex Composite Functional to Optimal Control in Variational Inequalities.- On Some Nondifferentiable Problems in Optimal Control.- On Sufficient Conditions for Optimality of Lipschitz Functions and Their Applications to Vector Optimization.- Optimal Control of Hyperbolic Variational Inequalities.- On Duality Theory Related to Approximate Solutions of Vector-Valued Optimization Problems.- III. Algorithms and Optimization Methods.- Seminormal Functions in Optimization Theory.- The General Concept of Cone Approximations in Nondifferentiable Optimization.- An Algorithm for Convex NDO Based on Properties of the Contour Lines of Convex Quadratic Functions.- A Note on the Complexity of an Algorithm for Tchebycheff Approximation.- Descent Methods for Nonsmooth Convex Constrained Minimization.- Stability Properties of Infima and Optimal Solutions of Parametric Optimization Problems.- On Methods for Solving Optimization Problems Without Using Derivatives.- An Accelerated Method for Minimizing a Convex Function of Two Variables.- On the Steepest-Descent Method for a Class of Quasi-Differentiable Optimization Problems.- A Modified Ellipsoid Method for the Minimization of Convex Functions With Superlinear Convergence (or Finite Termination) for Well-Conditioned C3 Smooth (or Piecewise Linear) Functions.- Numerical Methods for Multiextremal Nonlinear Programming Problems With Nonconvex Constraints.- A Modification of the Cutting-Plane Method With Accelerated Convergence.- A Finite Algorithm for Solving Linear Programs With an Additional Reverse Convex Constraint.- IV. Stochastic Programming and Applications.- Some Remarks on Quasi-Random Optimization.- Optimal Satellite Trajectories: a Source of Difficult Nonsmooth Optimization Problems.- A Reduced Subgradient Algorithm for Network Flow Problems With Convex Nondifferentiable Costs.- An Algorithm for Solving a Water-Pressure-Control Planning Problem With a Nondifferentiable Objective Function.- Quasi-Differentiable Functions in the Optimal Construction of Electrical Circuits.