CONVEXITY AND OPTIMIZATION IN RN

By C. FARAO

CONTENTS I TOPICS IN REAL ANALYSIS 1 INTRODUCTION 2 VECTORS IN R 3 ALGEBRA OF SETS 4 METRIC TOPOLOGY OF R 5 LIMITS AND CONTINUITY 6 BASIC PROPERTY OF REAL NUMBERS 7 COMPACTNESS 8 EQUIVALENT NORMS AND CARTESIAN PRODUCTS 9 FUNDAMENTAL EXISTENCE THEOREM 10 LINEAR TRANSFORMATIONS 11 DIFFERENTIATION IN R II CONVEX SETS IN RN 1 LINES AND HYPERPLANES IN R 2 PROPERTIES OF CONVEX SETS 3 SEPARATION THEOREMS 4 SUPPORTING HYPERPLANES: EXTREME POINTS 5 SYSTEMS OF LINEAR INEQUALITIES: THEOREMS OF THE ALTERNATIVE 6 AF?NE GEOMETRY 7 MORE ON SEPARATION AND SUPPORT III CONVEX FUNCTIONS 1 DE?NITION AND ELEMENTARY PROPERTIES 2 SUBGRADIENTS 3 DIFFERENTIABLE CONVEX FUNCTIONS 4 ALTERNATIVE THEOREMS FOR CONVEX FUNCTIONS 5 APPLICATION TO GAME THEORY IV OPTIMIZATION PROBLEMS 2 DIFFERENTIABLE UNCONSTRAINED PROBLEMS 3 OPTIMIZATION OF CONVEX FUNCTIONS 4 LINEAR PROGRAMMING PROBLEMS 5 FIRST-ORDER CONDITIONS FOR DIFFERENTIABLE NONLINEAR PROGRAMMING PROBLEMS 6 SECOND-ORDER CONDITIONS V CONVEX PROGRAMMING AND DUALITY 1 PROBLEM STATEMENT 2 NECESSARY CONDITIONS AND SUF?CIENT CONDITIONS 3 PERTURBATION THEORY 4 LAGRANGIAN DUALITY 5 GEOMETRIC INTERPRETATION 6 QUADRATIC PROGRAMMING 7 DUALITY IN LINEAR PROGRAMMING VI SIMPLEX METHOD 2 EXTREME POINTS OF FEASIBLE SET 3 PRELIMINARIES TO SIMPLEX METHOD 4 PHASE II OF SIMPLEX METHOD 5 TERMINATION AND CYCLING 6 PHASE I OF SIMPLEX METHOD 7 REVISED SIMPLEX METHOD