Miguel A. Goberna

Bio: Miguel A. Goberna is an academic researcher from University of Alicante. The author has contributed to research in topics: Convex analysis & Linear programming. The author has an hindex of 24, co-authored 136 publications receiving 2519 citations. Previous affiliations of Miguel A. Goberna include University of Valencia.

Linear Semi-Infinite Optimization

Abstract: MODELLING. Modelling with the Primal Problem. Modelling with the Dual Problem. LINEAR SEMI-INFINITE SYSTEMS. Alternative Theorems. Consistency. Geometry. Stability. THEORY OF LINEAR SEMI-INFINITE PROGRAMMING. Optimality. Duality. Extremality and Boundedness. Stability and Well-Posedness. METHODS OF LINEAR SEMI-INFINITE PROGRAMMING. Local Reduction and Discretization Methods. Simplex-Like and Exchange Methods. Appendix. Symbols and Abbreviations. References. Index.

New Farkas-type constraint qualifications in convex infinite programming

Abstract: 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.

Semi-infinite programming : recent advances

Abstract: 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.

Stability Theory for Linear Inequality Systems

Abstract: 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.

From linear to convex systems: consistency, Farkas' Lemma and applications

Abstract: This research was partially supported by MCYT of Spain and FEDER of EU, Grant BMF2002-04114- C02-01.

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

Abstract: 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.

Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations

Abstract: In this paper we consider (hierarchical, La-grange)reduced basis approximation anda posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equa-tions. The essential ingredients are (primal-dual)Galer-kin projection onto a low-dimensional space associated with a smooth “parametric manifold” - dimension re-duction; efficient and effective greedy sampling meth-ods for identification of optimal and numerically stable approximations - rapid convergence;a posteriori er-ror estimation procedures - rigorous and sharp bounds for the linear-functional outputs of interest; and Offine-Online computational decomposition strategies - min-imummarginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, control)and many-query (e.g., design optimization, multi-model/ scale)contexts. We present illustrative results for heat conduction and convection-diffusion,inviscid flow, and linear elasticity; outputs include transport rates, added mass,and stress intensity factors.

SET-Valued Analysis

Abstract: “Multivalued Analysis” is the theory of set-valued maps (called multifonctions) and has important applications in many different areas. Multivalued analysis is a remarkable mixture of many different parts of mathematics such as point-set topology, measure theory and nonlinear functional analysis. It is also closely related to “Nonsmooth Analysis” (Chapter 5) and in fact one of the main motivations behind the development of the theory, was in order to provide necessary analytical tools for the study of problems in nonsmooth analysis. It is not a coincidence that the development of the two fields coincide chronologically and follow parallel paths. Today multivalued analysis is a mature mathematical field with its own methods, techniques and applications that range from social and economic sciences to biological sciences and engineering. There is no doubt that a modern treatise on “Nonlinear Functional Analysis” can not afford the luxury of ignoring multivalued analysis. The omission of the theory of multifunctions will drastically limit the possible applications.