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Full Description
In this book, the author considers separable programming and, in particular, one of its important cases - convex separable programming. Some general results are presented, techniques of approximating the separable problem by linear programming and dynamic programming are considered.
Convex separable programs subject to inequality/ equality constraint(s) and bounds on variables are also studied and iterative algorithms of polynomial complexity are proposed.
As an application, these algorithms are used in the implementation of stochastic quasigradient methods to some separable stochastic programs. Numerical approximation with respect to I1 and I4 norms, as a convex separable nonsmooth unconstrained minimization problem, is considered as well.
Audience: Advanced undergraduate and graduate students, mathematical programming/ operations research specialists.
Contents
1 Preliminaries: Convex Analysis and Convex Programming.- One — Separable Programming.- 2 Introduction. Approximating the Separable Problem.- 3 Convex Separable Programming.- 4 Separable Programming: A Dynamic Programming Approach.- Two — Convex Separable Programming With Bounds On The Variables.- Statement of the Main Problem. Basic Result.- Version One: Linear Equality Constraints.- 7 The Algorithms.- 8 Version Two: Linear Constraint of the Form "?".- 9 Well-Posedness of Optimization Problems. On the Stability of the Set of Saddle Points of the Lagrangian.- 10 Extensions.- 11 Applications and Computational Experiments.- Three — Selected Supplementary Topics and Applications.- 12 Approximations with Respect to ?1 and ??-Norms: An Application of Convex Separable Unconstrained Nondifferentiable Optimization.- 13 About Projections in the Implementation of Stochastic Quasigradient Methods to Some Probabilistic Inventory Control Problems. The Stochastic Problem of Best Chebyshev Approximation.- 14 Integrality of the Knapsack Polytope.- Appendices.- A Appendix A — Some Definitions and Theorems from Calculus.- B Appendix B — Metric, Banach and Hilbert Spaces.- C Appendix C — Existence of Solutions to Optimization Problems — A General Approach.- D Appendix D — Best Approximation: Existence and Uniqueness.- Bibliography, Index, Notation, List of Statements.- Notation.- List of Statements.
