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Convex Analysisの二,三の進展について

Stochastic programming can effectively describe many decision-making problems in uncertain environments. Unfortunately, such programs are often computationally demanding to solve. In addition, their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model that describes uncertainty in both the distribution form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance matrix). We demonstrate that for a wide range of cost functions the associated distributionally robust (or min-max) stochastic program can be solved efficiently. Furthermore, by deriving a new confidence region for the mean and the covariance matrix of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. These arguments are confirmed in a practical example of portfolio selection, where our framework leads to better-performing policies on the “true” distribution underlying the daily returns of financial assets.

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Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems

Convex Analysis: (pms-28)

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

Supply chain collaboration: making sense of the strategy continuum

Semi-Infinite Programming

This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas' lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint qualifications and/or a certain closedness assumption.

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New Farkas-type constraint qualifications in convex infinite programming

Preface. Contributing Authors. Part I: History. 1. On the 1962-1972 Decade of Semi-Infinite Programming: A Subjective View K.O. Kortanek. Part II: Theory. 2. About Disjunctive Optimization I.I. Eremin. 3. On Regularity and Optimality in Nonlinear Semi-Infinite Programming A. Hassouni, W. Oettli. 4. Asymptotic Constraint Qualifications and Error Bounds for Semi-Infinite Systems of Convex Inequalities W. Li, I. Singer. 5. Stability of the Feasible Set Mapping in Convex Semi-Infinite Programming M.A. Lopez, et al. 6. On Convex Lower Level Problems in Generalized Semi-Infinite Optimization J.-J. Ruckmann, O. Stein. 7. On Duality Theory of Conic Linear Problems A. Shapiro. Part III: Numerical Methods. 8. Two Logarithmic Barrier Methods for Convex Semi-Infinite Problems L. Abbe. 9. First-Order Algorithms for Optimization Problems with a Maximum Eigenvalue/Singular Value Cost and or Constraints E. Polak. 10. Analytic Center Based Cutting Plane Method for Linear Semi-Infinite Programming S.-Y. Wu, et al. Part IV: Modeling and Applications. 11. On Some Applications of LSIP to Probability and Statistics M. Dall'Aglio. 12. Separation by Hyperplanes: A Linear Semi-Infinite Programming Approach M.A. Goberna, et al. 13. A Semi-Infinite Optimization Approach to Optimal Spline Trajectory Planning of Mechanical Manipulators C. Guarino Lo Bianco, A. Piazzi. 14. On Stability of Guaranteed Estimation Problems: Error Bounds for Information Domains and Experimental Design M.I. Gusev, S.A.Romanov. 15. Optimization under Uncertainty and Linear Semi-Infinite Programming: A Survey T. Leon, E. Vercher. 16. Semi-Infinite Assignment and Transportation Games J. Sanchez-Soriano, et al. 17. The Owen Set and the Core of Semi-Infinite Linear Production Situations S. Tijs, et al.

Semi-infinite programming : recent advances

This paper develops a stability theory for (possibly infinite) linear inequality systems defined on a finite-dimensional space, analyzing certain continuity properties of the solution set mapping. It also provides conditions under which sufficiently small perturbations of the data in a consistent (inconsistent) system produce systems belonging to the same class.

Stability Theory for Linear Inequality Systems

We provide a rule to calculate the subdifferential set of the pointwise supremum of an arbitrary family of convex functions defined on a real locally convex topological vector space. Our formula is given exclusively in terms of the data functions and does not require any assumption either on the index set on which the supremum is taken or on the involved functions. Some other calculus rules, namely chain rule formulas of standard type, are obtained from our main result via new and direct proofs.

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Marco A. López

Papers

Semi-Infinite Programming

New Farkas-type constraint qualifications in convex infinite programming

Semi-infinite programming : recent advances

Stability Theory for Linear Inequality Systems

Subdifferential Calculus Rules in Convex Analysis: A Unifying Approach Via Pointwise Supremum Functions