Schedule-at-a-Glance

34 PRIMA 2013 AbstractsDesign of robust truss structures for minimumweight using the sequential convex approximationmethodAlfredo CanelasUniversidad de la República, Uruguayacanelas@fing.edu.uyMiguel CarrascoUniversidad de los Andes, Chilemigucarr@uandes.clJulio LopezUniversidad Diego Portales, Chilejulio.lopez@udp.clTrusses are mechanical structures in R d , where d is 2 forplanar trusses or 3 for three-dimensional ones. They consistof an ensemble of N nodes joint by m ≥ 2 bars whichare made of a linear elastic, isotropic and homogeneousmaterial. Long bars overlapping small ones are not allowed,and therefore m ≤ N(N − 1)/2 for a mesh fullof bars. Trusses are designed to support some externalnodal loads taking into account the properties of the barmaterial. We assume that there exists a set of primaryexternal loads, applied only in the nodes of the truss, andthat there exists a set of secondary nodal ones that areuncertain in size and direction, which can be viewed asa perturbation of the main loads. Our main objective isto minimize the total amount of material or weight, withthe purpose of finding the most economic structure, thatis robust under load perturbations.We formulate a model in order to include the mechanicalequilibrium constraint and stress constraints, as well asbounds on displacements, considering also stability constraintsunder perturbation of the main load. More precisely,we will study the properties of the following non–convex semi-infinite mathematical programming problem(P w) minx≥0m∑x ii=1|u j (ξ, x)| ≤ ū j j ∈ J ⊆ {1, . . . , n},|σ i (ξ, x)| ≤ ¯σ i i ∈ I ⊆ {1, . . . , m},ξ ∈ E,where x i represents the volume of each bar, u is the vectorof displacements, and σ i the stress of the i-th bar. Theset E ⊆ R n corresponds to the set of secondary loads,and we implicitly assume the mechanical equilibrium constraintK(x)u(ξ, x) = f + ξ. To address the infinite numberof constraints we reformulate (P w) as a non–convexbilevel mathematical program. The main idea is to solvethis problem assuming that the set E of secondary loadstakes the form of a particular ellipsoid, similar to the approachedby Ben-Tal and Nemirovski . Based on the workof Beck et al., we have obtained encouraging preliminarynumerical results for this alternative, which approach consistson approximating some non–convex constraints andsolving a sequence of parametric conic problems.Expected residual minimization for stochasticvariational inequalitiesXiaojun ChenThe Hong Kong Polytechnic University, Hong Kong,Chinamaxjchen@polyu.edu.hkThis talk discusses a variety of computational approachesfor stochastic variational inequalities and presents a newexpected residual minimization formulation for a classof stochastic variational inequalities by using the gapfunction. The objective function of the expected residualminimization problem is nonnegative and Lipschitzcontinuous. Moreover, it is convex for some stochasticlinear variational inequalities, which helps us guaranteethe existence of a solution and convergence of approximationmethods. We propose a globally convergent(a.s.) smoothing sample average approximation (SSAA)method to minimize the expected residual function. Weshow that the residual minimization problem and itsSSAA problems have minimizers in a compact set andany cluster point of minimizers and stationary points ofthe SSAA problems is a minimizer and a stationary pointof the expected residual minimization problem (a.s.). Ourexamples come from applications involving traffic flowproblems. We show that the conditions we impose aresatisfied and that the solutions, efficiently generated bythe SSAA procedure, have desirable properties.Joint work with Roger Wets and Yangfan Zhang.Stability of two-stage stochastic programswith quadratic continuous recourseZhiping ChenXi’an Jiaotong Universityzchen@mail.xjtu.edu.cnYoupan HanXi’an Polytechnic UniversityWe consider the stability of the two-stage quadraticstochastic programming problem with quadratic continuousrecourse when the underlying probability distributionis perturbed. For the case that all the coefficientsin the objective function and the right-hand side vectorin second-stage constraints are random, we firstly showthe locally Lipschtiz continuity of the optimal value functionof the recourse problem; then under suitable probabilitymetric, we derive the Lipschitz continuity of theexpected optimal value function with respect to the firststagevariable and the probability distribution; and furthermore,we establish the qualitative and quantitativestability results of the optimal value function and theoptimal solution set with respect to the Fortet-Mourierprobability metric. For the more complex situation thatthe recourse costs, the recourse matrix, the technologymatrix and the right-hand side vector in the second-stageproblem are all random, and that the second-stage objectivefunction might be non-convex, we obtain the uniformlyboundedness and locally Lipschtiz continuity ofthe second-stage feasible solution set when it is regularand bounded; we also establish the locally Lipschtizcontinuity of the corresponding optimal value function;then we show the Lipschitz continuity of the expectedvalue function of the second-stage problem with respectto both the first-stage variables and the probability metricinvolved in the second-stage problem; utilizing theseresults, we finally derive the qualitative and quantitativestability results of the optimal value function andthe optimal solution set of two-stage quadratic stochasticprograms with respect to the Fortet-Mourier probabilitymetric.Existence of minimizers on dropsRafael CorreaUniversidad de Chile, Chilercorrea@dim.uchile.clFor a boundedly generated drop [a, E] (property whichholds, for instance, whenever E is bounded), where abelongs to a real Banach space X and E ⊂ X is anonempty convex set, we show that for every lower semicontinuousfunction h : X −→ R ∪ {+∞} that satisfiessup δ > 0 infx ∈ E + δBx h(x) > h(a)(B X being the unitaryopen ball in X), there exists x ∈ [a, E] such thath(a) ≥ h(x) and x is a strict minimizer of h on the drop[x, E].The robust stability of every equilibrium ineconomic models of exchangeAlejandro JofreUniversity de Chile, Chileajofre@dim.uchile.cl