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Mesh adaptive computation of upper and lower bounds in limit analysis

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2008年第8期75卷 899-944页

作者： Ciria, H. Peraire, J. Bonet, J. Swansea Univ Civil & Computat Engn Ctr Sch Engn Swansea SA2 8PP W Glam Wales MIT Aerosp Computat Design Lab Dept Aeronaut Cambridge MA 02139 USA

An efficient procedure to compute strict upper and lower bounds for the exact collapse multiplier in limit analysis is presented, with a formulation that explicitly considers the exact convex yield condition. The approach consists of two main steps. First, the continuous problem, under the form of the static principle of limit analysis, is discretized twice (one per bound) using particularly chosen finite element spaces for the stresses and velocities that guarantee the attainment of all upper or a lower bound. The second step consists of solving the resulting discrete non-linear optimization problems. These are reformulated as second-order cone programs, which allows for the use of primal-dual interior point methods that optimally exploit the convexity and duality properties of the limit analysis model. To benefit from the fact that collapse mechanisms are typically highly localized. a novel method for adaptive meshing is introduced. The method first decomposes the total bound gap as the sum of positive contributions from each element in the mesh and then refines those elements with higher contributions. The efficiency of the methodology is illustrated with applications in plane stress and plane strain problems. Copyright (C) 2008 John Wiley & Soils, Ltd.

关键词： limit analysis bounds conic programming mesh adaptivity finite elements

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A primal-dual symmetric relaxation for homogeneous conic systems

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JOURNAL OF COMPLEXITY 2007年第2期23卷 245-261页

作者： Vera, Juan Carlos Rivera, Juan Carlos Pena, Javier Hui, Yao Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15213 USA Carnegie Mellon Univ Dept Math Sci Pittsburgh PA 15213 USA City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China

We address the feasibility (existence of non-trivial solutions) of the pair of alternative conic systems of constraints Ax=0, x is an element of C and -A(T)y is an element of C8, where A is an element of R-mxn, m < n, is a full row-rank matrix, and C subset of R-n is a closed convex cone. To this end, we reformulate the above pair of conic systems as a primal-dual pair of conic programs. Each of the conic programs corresponds to a natural relaxation of each of the two conic systems. When C is a self-scaled cone with a known self-scaled barrier, the conic programming reformulation can be solved via an interior-point algorithm. For a well-posed instance A, a strict solution to one of the two original conic systems can be obtained in O(root nu(C) log(nu(C) C(A)) interior-point iterations. Here nu(C) is the complexity parameter of the self-scaled barrier of C and C(A) is Renegar's condition number of A. A central feature of our approach is the conditioning of the system of equations that arise at each interior-point iteration. The condition number of such system of equations grows in a controlled manner and remains bounded by a constant factor of C(A)(2) throughout the entire algorithm. (c) 2007 Elsevier Inc. All rights reserved.

关键词： conic programming condition numbers interior-point methods

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Full Nesterov-Todd step infeasible interior-point method for symmetric optimization

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2011年第3期214卷 473-484页

作者： Gu, G. Zangiabadi, M. Roos, C. Nanjing Univ Dept Math Nanjing Peoples R China Shahrekord Univ Dept Math Sci Shahrekord Iran Delft Univ Technol Fac Elect Engn Math & Comp Sci NL-2600 AA Delft Netherlands

Euclidean Jordan algebras were proved more than a decade ago to be an indispensable tool in the unified study of interior-point methods. By using it, we generalize the full-Newton step infeasible interior-point method for linear optimization of Roos [Roos, C., 2006. A full-Newton step O(n) infeasible interior-point algorithm for linear optimization. SIAM Journal on Optimization. 16(4), 1110-1136 (electronic)] to symmetric optimization. This unifies the analysis for linear, second-order cone and semidefinite optimizations. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Interior point methods conic programming Nesterov-Todd step

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LMI approximations for cones of positive semidefinite forms

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SIAM JOURNAL ON OPTIMIZATION 2006年第4期16卷 1076-1091页

作者： Zuluaga, LF Vera, J Peña, J Univ New Brunswick Fac Business Adm Fredericton NB E3B 5A3 Canada Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15213 USA Carnegie Mellon Univ Dept Math Sci Pittsburgh PA 15213 USA Univ Los Andes Bogota Colombia

An interesting recent trend in optimization is the application of semidefinite programming techniques to new classes of optimization problems. In particular, this trend has been successful in showing that under suitable circumstances, polynomial optimization problems can be approximated via a sequence of semidefinite programs. Similar ideas apply to conic optimization over the cone of copositive matrices and to certain optimization problems involving random variables with some known moment information. We bring together several of these approximation results by studying the approximability of cones of positive semidefinite forms ( homogeneous polynomials). Our approach enables us to extend the existing methodology to new approximation schemes. In particular, we derive a novel approximation to the cone of copositive forms, that is, the cone of forms that are positive semidefinite over the nonnegative orthant. The format of our construction can be extended to forms that are positive semidefinite over more general conic domains. We also construct polyhedral approximations to cones of positive semidefinite forms over a polyhedral domain. This opens the possibility of using linear programming technology in optimization problems over these cones.

关键词： positive polynomials global optimization linear matrix inequalities conic programming semidefinite programming

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SECOND-ORDER VARIATIONAL ANALYSIS OF PARAMETRIC CONSTRAINT AND VARIATIONAL SYSTEMS

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SIAM JOURNAL ON OPTIMIZATION 2019年第1期29卷 423-453页

作者： Gfrerer, Helmut Mordukhovich, Boris S. Johannes Kepler Univ Linz Inst Computat Math A-4040 Linz Austria Wayne State Univ Dept Math Detroit MI 48202 USA RUDN Univ Moscow 117198 Russia

This paper is devoted to the generalized differential study of the normal cone mappings associated with a large class of parametric constraint systems (PCS) that appear, in particular, in nonpolyhedral conic programming. Conducting a local second-order analysis of such systems, we focus on computing the (primal-dual) graphical derivative of the normal cone mapping under the C-2-cone reducibility of the constraint set together with the fairly weak metric subregularity constraint qualification and its uniform parametric counterpart known as Robinson stability. The obtained precise formulas for computing the underlying second-order object are applied to the derivation of comprehensive conditions ensuring the important stability property of isolated calmness for solution maps to parametric variational systems associated with the given PCS.

关键词： second-order variational analysis parametric constraint and variational systems normal cone mappings graphical derivatives conic programming C-2-cone reducibility metric subregularity

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Strong Stationarity for Optimization Problems with Complementarity Constraints in Absence of Polyhedricity With Applications to Optimization with Semidefinite and Second-Order-Cone Complementarity Constraints

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SET-VALUED AND VARIATIONAL ANALYSIS 2017年第1期25卷 133-175页

作者： Wachsmuth, Gerd Tech Univ Chemnitz Fac Math Professorship Numer Math Partial Differential Equ D-09107 Chemnitz Germany

We consider mathematical programs with complementarity constraints in Banach spaces. In particular, we focus on the situation that the complementarity constraint is defined by a non-polyhedric cone K. We demonstrate how strong stationarity conditions can be obtained in an abstract setting. These conditions and their verification can be made more precise in the case that Z is a Hilbert space and if the projection onto K is directionally differentiable with a derivative as given in Haraux (Journal of the Mathematical Society of Japan 29(4), 615-631, 1977, Theorem 1). Finally, we apply the theory to optimization problems with semidefinite and second-order-cone complementarity constraints. We obtain that local minimizers are strongly stationary under a variant of the linear-independence constraint qualification, and these are novel results.

关键词： Mathematical programs with complementarity constraints Strong stationarity conic programming Semidefinite cone Second-order cone Polyhedricity

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Constraint qualifications for convex inequality systems with applications in constrained optimization

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SIAM JOURNAL ON OPTIMIZATION 2008年第1期19卷 163-187页

作者： Li, Chong Ng, K. F. Pong, T. K. Zhejiang Univ Dept Math Hangzhou 310027 Peoples R China Chinese Univ Hong Kong Dept Math Hong Kong Hong Kong Peoples R China

For an inequality system defined by an infinite family of proper convex functions, we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions and study relationships between these new constraint qualifications and other well-known constraint qualifications including the basic constraint qualification studied by Hiriart-Urrutty and Lemarechal and by Li, Nahak, and Singer. Extensions of known results to more general settings are presented, and applications to particular important problems, such as conic programming and approximation theory, are also studied.

关键词： convex inequality system basic constraint qualification strong conical hull intersection property best constrained approximation conic programming

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Optimal Design of Plane Elastic Membranes Using the Convexified Föppl's Model

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APPLIED MATHEMATICS AND OPTIMIZATION 2024年第1期90卷 23-23页

作者： Bolbotowski, Karol Warsaw Univ Technol Fac Civil Engn Dept Struct Mech & Comp Aided Engn 16 Armii Ludowej St PL-00637 Warsaw Poland Lagrange Math & Comp Res Ctr 103 Rue Grenelle F-75007 Paris France

This work puts forth a new optimal design formulation for planar elastic membranes. The goal is to minimize the membrane's compliance through choosing the material distribution described by a positive Radon measure. The deformation of the membrane itself is governed by the convexified F & ouml;ppl's model. The uniqueness of this model lies in the convexity of its variational formulation despite the inherent nonlinearity of the strain-displacement relation. It makes it possible to rewrite the optimization problem as a pair of mutually dual convex variational problems. The primal variables are displacement functions, whilst in the dual one seeks stresses being Radon measures. The pair of problems is analysed: existence and regularity results are provided, together with the system of optimality criteria. To demonstrate the computational potential of the pair, a finite element scheme is developed around it. Upon reformulation to a conic-quadratic & semi-definite programming problem, the method is employed to produce numerical simulations for several load case scenarios.

关键词： Plane membrane Optimal design Convexified F & ouml ppl's model Convex duality conic programming

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Mixed integer programming with a class of nonlinear convex constraints

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DISCRETE OPTIMIZATION 2017年 24卷 66-86页

作者： Vinel, Alexander Krokhmal, Pavlo A. Auburn Univ Dept Ind & Syst Engn Shelby Ctr 3301 Auburn AL 36849 USA Univ Arizona Dept Syst & Ind Engn 1127 E James E Rogers Way Tucson AZ 85721 USA

We study solution approaches to a class of mixed-integer nonlinear programming problems that arise from recent developments in risk-averse stochastic optimization and contain second- and p-order cone programming as special cases. We explore possible applications of some of the solution techniques that have been successfully used in mixed-integer conic programming and show how they can be generalized to the problems under consideration. Particularly, we consider a branch-and-bound method based on outer polyhedral approximations, lifted nonlinear cuts, and linear disjunctive cuts. Results of numerical experiments with discrete portfolio optimization models are presented. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Mixed-integer nonlinear programming Measures of risk Quasi-arithmetic average Valid inequalities conic programming

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Return-mapping algorithms for associative isotropic hardening plasticity using conic optimization

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APPLIED MATHEMATICAL MODELLING 2020年 78卷 724-748页

作者： Bruno, Hugo Barros, Guilherme Menezes, Ivan F. M. Martha, Luiz Fernando Pontificia Univ Catolica Rio de Janeiro Rua Marques de Sao Vicente 225 BR-22451900 Rio De Janeiro RJ Brazil

We present a mathematical programming approach for elastoplastic constitutive initial value problems. Consideration of the associative plasticity and a linear isotropic hardening model allowed us to formulate the local discrete constitutive equations as conic programs. Specifically, we demonstrate that implicit return-mapping schemes for well-known yield criteria, such as the Rankine, von Mises, Tresca, Drucker-Prager, and Mohr-Coulomb criteria, can be expressed as second-order and semidefinite conic programs. Additionally, we propose a novel scheme for the numerical evaluation of the consistent elastoplastic tangent operator based on a first-order parameter derivative of the optimal solutions. (C) 2019 Elsevier Inc. All rights reserved.

关键词： Return-Mapping algorithms Elastoplastic analysis Implicit Euler scheme conic programming First-Order parameter derivatives

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