Semi-Infinite Programming and Applications

Semi-infinite programming is a natural extension of linear pro­ gramming that allows finitely many variables to appear in infinitely many constraints. As the papers in this collection will reconfirm, the theoretical and practical manifestations and applications of this prob­ lem formulation are abundant and significant. This volume presents 20 carefully selected papers that were pre­ sented at the International Symposium on Semi-Infinite Programming and Applications, The University of Texas at Austin, September 8-10, 1981. A total of 70 papers were presented by distinguished participants from 15 countries. This was only the second international meeting on this topic, the first taking place in Bad Honnef,Federal Republic of Germany in 1978. A proceedings of that conference was organized and edited by Rainer Hettich of the University of Trier and published by Springer­ Verlag in 1979. The papers in this volume could have been published in any of several refereed journals. It is also probable that the authors of these papers would normally not have met at the same professional society meeting. Having these papers appear under one cover is thus something of a new phenomenon and provides an indication of both the unification and cross-fertilization opportunities that have emerged in this field. These papers were solicited only through the collective efforts of an International Program Committee organized according to the fol­ lowing research areas.

I Duality Theory.- Ascent Ray Theorems and Some Applications.- 1. Introduction.- 2. Application to Semi-Infinite Programming.- 3. Intersections of Convex Sets.- 4. Convex Optimization Application.- 5. What About Systems With Infinitely Many Variables?.- References.- Semi-Infinite Programming Duality: How Special Is It?.- 1. Introduction.- 2. Direct Techniques: (I) Primal Reduction Theorems.- 3. Direct Techniques: (II) Dual Reduction Theorems.- 4. Abstract Linear Duality.- 5. Specializations.- 6. Abstract Convex and Differentiable Programming.- 7. Nonsmooth Techniques.- 8. Conclusion.- References.- A Saddle Value Characterization of Fan’s Equilibrium Points.- 1. Introduction: A Biextremal Formulation.- 2. A Separably-Infinite, Biextremal Formulation of the Fan Equilibrium Problem.- 3. Equivalent Dual Pair of Separably-Infinite Programs.- 4. The Fan Equilibrium as the Unique Zero of VM(?).- 5. The Fan Equilibrium Value as Saddle Value of a Ratio Game.- 6. Conclusion.- References.- Duality in Semi-Infinite Linear Programming.- 1. Introduction.- 2. The Homogeneous Case.- 3. The Inhomogeneous Case and Duality Results.- References.- On the Role of Duality in the Theory of Moments.- 1. Introduction.- 2. A General Moment Problem.- 3. The Finite Case.- 4. A General Result.- 5. A General Transportation Problem.- 6. Theorems of Kantorovich, Rubinstein, Nachbin, and Strassen.- 7. Transshipment.- 8. Appendix.- References.- Existence Theorems in Semi-Infinite Programs.- 1. Introduction with Problem Setting.- 2. Existence Theorems for NSIP.- 3. Semi-Infinite Quadratic Program.- References.- II Algorithmic Developments.- An Algorithm for a Continuous Version of the Assignment Problem.- 1. The Continuous Transportation Problem.- 2. Basic Solutions and Assignments.- 3. The Continuous Assignment Problem and Its Algorithm.- References.- Numerical Estimation of Optima by Use of Duality Inequalities.- 1. Introduction.- 2. Minima of Functions of One Variable.- 3. Minima of Functions of Two Variables.- 4. The Apex Program.- 5. The Question of a Duality Gap.- 6. Estimating the Coordinates of an Optimum Point.- 7. Estimating Constrained Minima.- 8. Discussion.- References.- Globalization of Locally Convergent Algorithms for Nonlinear Optimization Problems with Constraints.- 1. Introduction.- 2.1. Local Stability Sets, Critical Points.- 2.2. Convergence of Local Methods on Local Stability Sets.- 2.3. Determination of Critical Points.- 2.4. Determination of the New Active Index Set.- 3. A Concept of a Globally Convergent Algorithm.- 4. A Concrete Imbedding for Convex Optimization Problems.- References.- A Three-Phase Algorithm for Semi-Infinite Programs.- 1. Introduction.- 2. Semi-Infinite Programs of P-type.- 3. Semi-Infinite Programs of D-type.- 4. Necessary Condition for Optimality.- 5. Approximation of Programs (P) and (D) with Discretized Problems.- 6. A General Three-Phase Algorithm.- References.- A Review of Numerical Methods for Semi-Infinite Optimization.- 1. Introduction.- 2. Exchange Methods.- 3. Linear Semi-Infinite and Differentiable Convex Programming.- 4. The Relation of Exchange Methods to Cutting Plane Methods.- 5. The Case of a Strongly Unique Solution.- 6. A Discretization Method.- 7. Local Reduction to a Finite Convex Problem.- 8. Some Examples of Methods for Solving the Reduced Problem.- 9. Some Remarks on the Nonlinear Case.- References.- An Algorithm for Minimizing Polyhedral Convex Functions.- 1. Introduction.- 2. A Continuation Procedure.- 3. Treatment of Degeneracy.- 4. Points Relating to Computation.- References.- Numerical Experiments with Globally Convergent Methods for Semi-Infinite Programming Problems.- 1. Introduction.- 2. A Model Algorithm.- 3. An Implementation of the Algorithm.- 4. Numerical Results.- 5. A Modified Subproblem.- 6. Concluding Remarks.- References.- III Problem Analysis and Modeling.- On the Partial Construction of the Semi-Infinite Banzhaf Polyhedron.- 1. Introduction.- 2. The Semi-Infinite Problem.- 3. Properties of the Semi-Infinite Polyhedron.- 4. Optimization.- References.- Semi-Infinite and Fuzzy Set Programming.- 1. Introduction.- 2. C/C Semi-Infinite Linear Programs.- 3. Fuzzy Set Programming.- 4. Relationship with C/C Semi-Infinite Programs.- References.- Semi-Infinite Optimization in Engineering Design.- 1. Introduction.- 2. Formulation of Engineering Design Problems in SIP Form.- 2.1. Seismic Resistant Design of Structures.- 2.2. Design of SISO Control Systems.- 2.3. Design of MIMO Control Systems.- 2.4. Electronic Circuit Design.- 3. SIP Algorithms for Engineering Design.- 4. Conclusion.- References.- A Moment Inequality and Monotonicity of an Algorithm.- 1. Introduction.- 2. A Problem.- 3. Examples.- 4. Sufficient Conditions for Optimality — Two Alternative Forms.- 5. Algorithms.- 6. A First or Intermediate Phase Algorithm.- 7. Moment Lemma and a Sufficient Condition.- 8. Empirical Information.- References.- IV Optimality Conditions and Variational Principles.- Second Order Conditions in Nonlinear Nonsmooth Problems of Semi-Infinite Programming.- 1. Introduction.- 2. Statements of Main Results.- 3. An Auxiliary Problem.- 4. Proofs of Main Theorems.- References.- On Stochastic Control Problems with Impulse Cost Vanishing.- 1. Introduction.- 2. Assumptions and Notations.- 3. Stochastic Impulse Control Problem.- 3.1. The General Case.- 3.2. A Particular Case.- 4. Existence of An Optimal Impulse Control.- References.- Dual Variational Principles in Mechanics and Physics.- 1. Description of the Primal Problem — Examples.- 2. Dual Problem.- 3. Relaxed Problem and Extension of Duality.- References.- Authors, Participants, and Affiliations.- Referees.- Table of Contents of the Book of Abstracts.