Schedule-at-a-Glance

35 PRIMA 2013 AbstractsIn an economic model of exchange of goods, the structurecan be specified by utility functions. Under utility conditionsidentified here even more broadly than usual, exceptfor concavity in place of quasi-concavity, every equilibriumwill simultaneously be stable with respect to shiftsin the associated holdings of the agents and with respectto the dynamic Walrasian tatonnement process of priceadjustment. This fact, seemingly contrary to widespreadbelief, is revealed by paying close attention not only toprices but also to the proximal status of initial holdings.The conditions on the concave utility functions are standardfor stability investigations, in that they invoke propertiescoming from second derivatives, but significantlyrelaxed in not forcing all goods to be held only in positiveamounts. Recent advances in variational analysisprovide the support needed for working in that context.The stability results also point the way toward furtherdevelopments in which an equilibrium might evolve inresponse to exogenous inputs to the agents’ holdings, orextractions.Solving mathematical programs with equilibriumconstraints as constrained equationsGui-Hua LinShanghai University, Chinaguihualin@shu.edu.cnLei GuoDalian University of Technology, Chinaguolayne@gmail.comJane J. YeUniversity of Victoria, Canadajaneye@uvic.caThis paper aims at developing effective numerical methodsfor solving mathematical programs with equilibriumconstraints. Due to the complementarity constraints, theusual constraint qualifications such as the Mangasarian-Fromovitz constraint qualification do not hold at any feasiblepoint and there are various stationarity conceptssuch as Clarke/Mordukhovich/strong stationarity suggestedin the literature. In this paper, we reformulatethese stationarity conditions as smooth equations withbox constraints. We then present a modified Levenberg-Marquardt method for solving these constrained equations.We show that, under some weak local error boundconditions, the method is locally and superlinearly convergent.Furthermore, we give some sufficient conditionsfor local error bounds to hold and show that these conditionsare not very stringent by a number of examples.A feasible direction algorithm for nonlinearsecond-order cone optimization problemsJulio LópezUniversidad Diego Portales, Chilejulio.lopez@udp.clIn this work, we propose a feasible direction algorithmfor solving nonlinear convex second-order cone programs.This kind of problems consists of minimizing a convexfunction over the Cartesian product of second-ordercones. Our approach computes feasible descend directionsby using the same formulation as in FDIPA proposed byHerskovitz [J. Optim. Theory Appl., vol. 99 (1998), pp.53–58]. Under suitable assumptions, some results preliminariesof convergence are obtained. Finally, to show howour algorithm works in practice, computational resultsapplied to support vector machines under uncertainty ispresented.A dynamical approach to an inertial forwardbackwardalgorithm for convex minimizationJuan PeypouquetUniversidad Técnica Federico Santa María, Chilejuan.peypouquet@usm.clWe introduce a new class of forward-backward algorithmsfor structured convex minimization problems in Hilbertspaces. Our approach relies on the time discretization ofa second-order differential system with two potentials andHessian-driven damping. This system can be equivalentlywritten as a first order-system in time and space, each ofthe two constitutive equations involving only one of thetwo potentials. Its time dicretization naturally leads tothe introduction of forward-backward splitting algorithmswith inertial features. Using a Liapunov analysis, weshow the convergence of the algorithm under conditionsimproving the classical step size limitation. Then, wespecialize our results to gradient-projection algorithms,and give some illustration to sparse signal recovery andfeasibility problems.Second-order analysis in conic programming:Applications to stabilityHéctor Ramírez C.Universidad de Chilehramirez@dim.uchile.clIn this talk we review some recent results obtained forconic programs from the application of a second-ordergeneralized differential approach. It is used to calculateappropriate derivatives and coderivatives of the correspondingsolution maps. These developments allow usto obtain verifiable conditions for the strong regularity,the Aubin property and isolated calmness of the consideredsolution maps. The main results obtained in thegeneral conic programming setting are specified for andillustrated by the second-order cone programming[1] J. F. Bonnans and H. Ramírez C., Perturbation analysisof second-order cone programming problems. Math.Program., 104 (2005), pp. 205-227.[2] J. V. Outrata and H. Ramírez C., On the Aubin propertyof critical points to perturbed second-order cone programs,SIAM J. Optim., 21 (2011), pp. 798-823.[3] B. Mordukhovich, J. V. Outrata and H. Ramírez C.,Second-order variational analysis in conic programmingwith application to optimality and stability. Submitted(2013).Facially exposed cones are not always niceVera RoshchinaUniversity of Ballarat, Australiavroshchina@ballarat.edu.auWe show that Gabor Pataki’s conjecture that facially exposedcones are nice is true in three-dimensional case,but does not hold for higher dimensions by presenting afour-dimensional counterexample. We also discuss implicationsof this result and state some open problems.Asymptotic convergence analysis for distributionalrobust optimization and equilibriumproblemsHailin SunHarbin Institute of Technology, Chinamathhlsun@gmail.comHuifu XuCity University of London, UKHuifu.Xu.1@city.ac.ukIn this paper, we study distributional robust optimizationapproaches for a one stage stochastic minimization problem,where the true distribution of the underlying randomvariables is unknown but it is possible to constructa set of probability distributions which contains the truedistribution and optimal decision is taken on the basis ofworst possible distribution from that set. We consider thecase when the distributional set is constructed through