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21
Applications of Secondorder Cone Programming
, 1998
"... In a secondorder cone program (SOCP) a linear function is minimized over the intersection of an affine set and the product of secondorder (quadratic) cones. SOCPs are nonlinear convex problems that include linear and (convex) quadratic programs as special cases, but are less general than semidefin ..."
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Cited by 216 (10 self)
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In a secondorder cone program (SOCP) a linear function is minimized over the intersection of an affine set and the product of secondorder (quadratic) cones. SOCPs are nonlinear convex problems that include linear and (convex) quadratic programs as special cases, but are less general than semidefinite programs (SDPs). Several efficient primaldual interiorpoint methods for SOCP have been developed in the last few years. After reviewing
On implementing a primaldual interiorpoint method for conic quadratic optimization
 MATHEMATICAL PROGRAMMING SER. B
, 2000
"... Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linea ..."
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Cited by 75 (6 self)
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Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic quadratic optimization problems can in theory be solved efficiently using interiorpoint methods. In particular it has been shown by Nesterov and Todd that primaldual interiorpoint methods developed for linear optimization can be generalized to the conic quadratic case while maintaining their efficiency. Therefore, based on the work of Nesterov and Todd, we discuss an implementation of a primaldual interiorpoint method for solution of largescale sparse conic quadratic optimization problems. The main features of the implementation are it is based on a homogeneous and selfdual model, handles the rotated quadratic cone directly, employs a Mehrotra type predictorcorrector
An Efficient PrimalDual InteriorPoint Method for Minimizing a Sum of Euclidean Norms
 SIAM J. SCI. COMPUT
, 1998
"... The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many different kinds of modern scientific applications. We derive a primaldual inte ..."
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Cited by 31 (1 self)
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The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many different kinds of modern scientific applications. We derive a primaldual interiorpoint algorithm for the problem, by applying Newton's method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition. The main work at each step consists of solving a system of linear equations (the Schur complement equations). This Schur complement matrix is not symmetric, unlike in linear programming. We incorporate a Mehrotratype predictorcorrector scheme and present some experimental results comparing several variations of the algorithm, including, as one option, explicit symmetrization of the Schur complement with a skew corrector term. We also present results obtained from a code implemented to so...
A Newton Barrier method for Minimizing a Sum of Euclidean Norms subject to linear equality constraints
, 1995
"... An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN ). The linear equality constraints are handled using an e ..."
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Cited by 27 (2 self)
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An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN ). The linear equality constraints are handled using an exact L 1 penalty function which is made smooth in the same way as the Euclidean norms. It is shown that the dual problem is to maximize a linear objective function subject to homogeneous linear equality constraints and quadratic inequalities. Hence the suggested method also solves such problems efficiently. In fact such a problem from plastic collapse analysis motivated this work. Numerical results are presented for large sparse problems, demonstrating the extreme efficiency of the method. Keywords: Sum of Norms, Nonsmooth Optimization, Duality, Newton Barrier Method. AMS(MOS) subject classification: 65K05, 90C06, 90C25, 90C90. Abbreviated title: A Newton barrier method. Supported by the ...
Mesh adaptive computation of upper and lower bounds in limit analysis
"... 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 t ..."
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Cited by 6 (1 self)
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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 an upper or a lower bound. The second step consists of solving the resulting discrete nonlinear optimization problems. These are reformulated as secondorder 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 q 2008 John Wiley
Automatic Mesh Refinement in Limit Analysis
, 1999
"... A strategy for automatic mesh refinement in limit analysis is combined with a recently developed computational method. In the absence of estimates of the local error the strategy can be based on the deformations and on slack in the yield condition. The approach is tested on standard problems in plan ..."
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Cited by 3 (0 self)
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A strategy for automatic mesh refinement in limit analysis is combined with a recently developed computational method. In the absence of estimates of the local error the strategy can be based on the deformations and on slack in the yield condition. The approach is tested on standard problems in plane strain, including the classical punch problem. Very accurate results are obtained with the use of moderate computational power. Key words: Limit analysis, plasticity, finite element method, automatic mesh refinement. AMS(MOS)subject classifications: 65N30, 65N50, 73E20, 90C90. Abbreviated title: Mesh refinement in limit analysis. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark. (edc@imada.sdu.dk) y KMD Odense, Odense, Denmark. (ols@kmd.dk) 1 Introduction During the past decade the development of convex nonlinear optimization methods has made it possible to solve the collapse problem of limit analysis on a large scale. In [ACO98] and...
Robustness analysis of structures based on plastic limit analysis with uncertain loads
 Journal of Mechanics of Materials and Structures
, 2008
"... This paper presents a method for computing an infogap robustness function of structures, which is regarded as one measure of structural robustness, under uncertainties associated with the limit load factor. We assume that the external load in the plastic limit analysis is uncertain around its nomin ..."
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Cited by 2 (1 self)
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This paper presents a method for computing an infogap robustness function of structures, which is regarded as one measure of structural robustness, under uncertainties associated with the limit load factor. We assume that the external load in the plastic limit analysis is uncertain around its nominal value. Various uncertainties are considered for the live, dead, and reference disturbance loads based on the nonstochastic infogap uncertainty model. Although the robustness function is originally defined by using the optimization problem with infinitely many constraints, we show that the robustness function is obtained as an optimal value of a linear programming (LP) problem. Hence, we can easily compute the infogap robustness function associated with the limit load factor by solving an LP problem. As the second contribution, we show that the robust structural optimization problems of trusses and frames can also be reduced to LP problems. In numerical examples, the robustness functions, as well as the robust optimal designs, are computed for trusses and framed structures by solving LP problems. 1.
CBLIB 2014: A benchmark library for conic mixedinteger and continuous optimization. Optimization Online
, 2014
"... Abstract The Conic Benchmark Library (CBLIB 2014) is a collection of more than a hundred conic optimization instances under a free and open license policy. It is the first extensive benchmark library for the advancing field of conic mixedinteger and continuous optimization, which is already suppor ..."
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Abstract The Conic Benchmark Library (CBLIB 2014) is a collection of more than a hundred conic optimization instances under a free and open license policy. It is the first extensive benchmark library for the advancing field of conic mixedinteger and continuous optimization, which is already supported by all major commercial solvers and spans a wide range of industrial applications. The library addresses the particular need for public test sets mixing cone types and allowing integer variables, but has all types of conic optimization in target. The CBF file format is embraced as standard, and tools are provided to aid integration with, or transformation to the input format of, any software package.
pp. X–XX NUMERICAL SOLUTION OF A VARIATIONAL PROBLEM ARISING IN STRESS ANALYSIS: THE VECTOR CASE
"... (Communicated by the associate editor name) Dedicated to Professor Roger Temam on the occasion of his 70th birthday Abstract. In this article, we discuss the numerical solution of a constrained minimization problem arising from the stress analysis of elastoplastic bodies. This minimization problem ..."
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(Communicated by the associate editor name) Dedicated to Professor Roger Temam on the occasion of his 70th birthday Abstract. In this article, we discuss the numerical solution of a constrained minimization problem arising from the stress analysis of elastoplastic bodies. This minimization problem has the flavor of a generalized nonsmooth eigenvalue problem, with the smallest eigenvalue corresponding to the load capacity ratio of the elastic body under consideration. An augmented Lagrangian method, together with finite element approximations, is proposed for the computation of the optimum of the nonsmooth objective function, and the corresponding minimizer. The augmented Lagrangian approach allows the decoupling of some of the nonlinearities and of the differential operators. Similarly an appropriate Lagrangian functional, and associated Uzawa algorithm with projection, are introduced to treat nonsmooth equality constraints. Numerical results validate the proposed methodology for various twodimensional geometries. 1. Introduction and Motivations