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Monotonicity of primaldual interiorpoint algorithms for semidefinite programming problems
, 1998
"... We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly imp ..."
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Cited by 218 (36 self)
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We present primaldual interiorpoint algorithms with polynomial iteration bounds to find approximate solutions of semidefinite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.
Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 84 (2 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
Robust Truss Topology Design via Semidefinite Programming
 OPTIMIZATION LABORATORY, FACULTY OF INDUSTRIAL ENGINEERING AND MANAGEMENT, TECHNION – THE ISRAEL INSTITUTE OF TECHNOLOGY, TECHNION CITY, HAIFA 32000
, 1995
"... We present and motivate a new model of the Truss Topology Design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small “occasional” loads is optimized. It is shown that the resulting optimization problem is a Semidefinite Program. We derive and ana ..."
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Cited by 65 (9 self)
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We present and motivate a new model of the Truss Topology Design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small “occasional” loads is optimized. It is shown that the resulting optimization problem is a Semidefinite Program. We derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples.
Strong duality for semidefinite programming
 SIAM J. Optim
, 1997
"... Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interiorpoint methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite ..."
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Cited by 63 (18 self)
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Abstract. It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interiorpoint methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap. We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.
SelfScaled Cones and InteriorPoint Methods in Nonlinear Programming
 Working Paper, CORE, Catholic University of Louvain, LouvainlaNeuve
, 1994
"... : This paper provides a theoretical foundation for efficient interiorpoint algorithms for nonlinear programming problems expressed in conic form, when the cone and its associated barrier are selfscaled. For such problems we devise longstep and symmetric primaldual methods. Because of the special ..."
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Cited by 30 (2 self)
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: This paper provides a theoretical foundation for efficient interiorpoint algorithms for nonlinear programming problems expressed in conic form, when the cone and its associated barrier are selfscaled. For such problems we devise longstep and symmetric primaldual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier. Key words: Nonlinear Programming, conical form, interior point algorithms, selfconcordant barrier, selfscaled cone, selfscaled barrier, pathfollowing algorithms, potentialreduction algorithms. AMS 1980 subject classification. Primary: 90C05, 90C25, 65Y20. CORE, Catholic University of Louvain, LouvainlaNeuve, Belgium. Email: nesterov@core.ucl.ac.be. Part of this work was done while the author was visiting the Cornell C...
A Computational Study of the Homogeneous Algorithm for LargeScale Convex Optimization
, 1997
"... Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of th ..."
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Cited by 23 (2 self)
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Recently the authors have proposed a homogeneous and selfdual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interiorpoint type method, nevertheless it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems that also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for largescale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark. Email: eda@busieco.ou.dk y ...
Topology Optimization Of Sheets In Contact By A Subgradient Method
 International Journal of Numerical Methods in Engineering
, 1997
"... . We consider the solution of finite element discretized optimum sheet problems by an iterative algorithm. The problem is that of maximizing the stiffness of a sheet subject to constraints on the admissible designs and unilateral contact conditions on the displacements. The model allows for zero des ..."
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Cited by 12 (8 self)
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. We consider the solution of finite element discretized optimum sheet problems by an iterative algorithm. The problem is that of maximizing the stiffness of a sheet subject to constraints on the admissible designs and unilateral contact conditions on the displacements. The model allows for zero design volumes, and thus constitutes a true topology optimization problem. We propose and evaluate a subgradient optimization algorithm for a reformulation into a nondifferentiable, convex minimization problem in the displacement variables. The convergence of this method is proven, and its low computational complexity is established. An optimal design is derived through a simple averaging scheme which combines the solutions to the linear design problems solved within the subgradient method. To illustrate the efficiency of the algorithm and investigate the properties of the optimal designs, the algorithm is numerically tested on some medium and large scale problems. Key words: optimum sheet; un...
SemiDefinite Problems in Truss Topology Optimization
, 1995
"... In this report we review optimization problems arising from truss topology design, which can be formulated as positive semidefinite problems (PSP's). This is done with a view towards applying primaldual interior point methods for PSP's to obtain efficient nonlinear solvers for this class ..."
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Cited by 5 (2 self)
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In this report we review optimization problems arising from truss topology design, which can be formulated as positive semidefinite problems (PSP's). This is done with a view towards applying primaldual interior point methods for PSP's to obtain efficient nonlinear solvers for this class of problems.
A.: Structural Design via Semidefinite Programming. In
 Vandenberghe (Eds.): Handbook on Semidefinite Programming
, 2000
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A Subgradient Method For Contact Structural Optimization
 In: Complementarity and Variational ProblemsState of the Art, Proceedings of the International Conference on Complementarity Problems (ICCP95
, 1995
"... . We consider the problem of maximizing the stiffness of a structure in unilateral contact by using a subgradient optimization algorithm. The problem is in general a nonlinear programming problem subject to equilibrium constraints, but can in certain cases be given a convex concave saddle point fo ..."
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Cited by 3 (3 self)
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. We consider the problem of maximizing the stiffness of a structure in unilateral contact by using a subgradient optimization algorithm. The problem is in general a nonlinear programming problem subject to equilibrium constraints, but can in certain cases be given a convex concave saddle point formulation. The structure is assumed to have a linear potential energydesign dependence, an assumption which is fulfilled by discretized variable thickness sheets and trusses. The subgradient optimization scheme is applied to the primal (displacementonly) reformulation of the saddle problem; convergence is guaranteed when design variables are allowed to take zero values, and hence topology optimization is included. The algorithm is augmented with a simple averaging scheme for generating an optimal design as well. We establish a finite termination criterion for the subgradient algorithm, and evaluate the numerical properties of the algorithm and its viability for solving largescale enginee...