Results 1  10
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21
Semidefinite optimization
 Acta Numerica
, 2001
"... Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the ..."
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Cited by 107 (2 self)
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Optimization problems in which the variable is not a vector but a symmetric matrix which is required to be positive semidefinite have been intensely studied in the last ten years. Part of the reason for the interest stems from the applicability of such problems to such diverse areas as designing the strongest column, checking the stability of a differential inclusion, and obtaining tight bounds for hard combinatorial optimization problems. Part also derives from great advances in our ability to solve such problems efficiently in theory and in practice (perhaps “or ” would be more appropriate: the most effective computational methods are not always provably efficient in theory, and vice versa). Here we describe this class of optimization problems, give a number of examples demonstrating its significance, outline its duality theory, and discuss algorithms for solving such problems.
InfeasibleStart PrimalDual Methods And Infeasibility Detectors For Nonlinear Programming Problems
 Mathematical Programming
, 1996
"... In this paper we present several "infeasiblestart" pathfollowing and potentialreduction primaldual interiorpoint methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a selfdual homogeneous primaldual problem. The methods under considerat ..."
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Cited by 31 (5 self)
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In this paper we present several "infeasiblestart" pathfollowing and potentialreduction primaldual interiorpoint methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a selfdual homogeneous primaldual problem. The methods under consideration generate an fflsolution for an fflperturbation of an initial strictly (primal and dual) feasible problem in O( p ln fflae f ) iterations, where is the parameter of a selfconcordant barrier for the cone, ffl is a relative accuracy and ae f is a feasibility measure. We also discuss the behavior of pathfollowing methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O( p ln ae \Delta ) iterations, where ae \Delta is a primal or dual infeasibility measure. 1 Introduction Nesterov and Nemirovskii [9] first developed and investigated extensions of several classes of interiorpoint algorithms for linear programming t...
Conic Convex Programming And SelfDual Embedding
 Optim. Methods Softw
, 1998
"... How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big M technique. In the interior point method, there is a more ..."
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Cited by 18 (2 self)
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How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the selfdual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded selfdual problem. The embedded selfdual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the socalled stri...
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 13 (1 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 ...
Polynomiality of PrimalDual Affine Scaling Algorithms for Nonlinear Complementarity Problems
, 1995
"... This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappi ..."
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Cited by 10 (4 self)
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This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primaldual affine scaling algorithms generates an approximate solution (given a precision ffl) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial of n, ln(1=ffl) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in [13].
A New SelfDual Embedding Method for Convex Programming
 Journal of Global Optimization
, 2001
"... In this paper we introduce a conic optimization formulation for inequalityconstrained convex programming, and propose a selfdual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint function ..."
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Cited by 9 (2 self)
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In this paper we introduce a conic optimization formulation for inequalityconstrained convex programming, and propose a selfdual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primaldual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the pathfollowing procedure, we may apply the selfconcordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed selfconcordant when the original constraint functions are convex and quadratic. Keywords: Convex Programming, Convex Cones, SelfDual Embedding, SelfConcordant Barrier Functions. # Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported by Hong Kong RGC Earmarked Grants CUHK4181/00E and CUHK4233/01E. 1 1
Solving Semidefinite Programs using Preconditioned Conjugate Gradients
 Optim. Methods Softw
, 2003
"... The contribution of this paper is to describe a general technique to solve some classes of large but sparse semidefinite problems via a robust primaldual interiorpoint technique which uses an inexact GaussNewton approach with a matrix free preconditioned conjugate gradient method. This approach a ..."
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Cited by 8 (3 self)
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The contribution of this paper is to describe a general technique to solve some classes of large but sparse semidefinite problems via a robust primaldual interiorpoint technique which uses an inexact GaussNewton approach with a matrix free preconditioned conjugate gradient method. This approach avoids the illconditioning pitfalls that result from symmetrization and from forming the socalled normal equations, while maintaining the primaldual framework.
Interiorpoint methods for optimization
, 2008
"... This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twen ..."
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Cited by 6 (0 self)
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This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.
Method of Approximate Centers for SemiDefinite Programming
, 1996
"... The success of interior point algorithms for largescale linear programming has prompted researchers to extend these algorithms to the semidefinite programming (SDP) case. In this paper, the method of approximate centers of Roos and Vial [13] is extended to SDP. Key words: Semidefinite programmi ..."
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Cited by 3 (3 self)
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The success of interior point algorithms for largescale linear programming has prompted researchers to extend these algorithms to the semidefinite programming (SDP) case. In this paper, the method of approximate centers of Roos and Vial [13] is extended to SDP. Key words: Semidefinite programming, pathfollowing algorithm, approximate centers Running title: Method of approximate centers for semidefinite programming. iii 1 Introduction In recent years, a great revival of interest in semidefinite programming (SDP) has taken place. The reason is basically twofold: Important applications in control theory [4], structural optimization [3], and combinatorial optimization [1], to name but a few, have been formulated as SDP problems. (A review of applications may be found in [15]). Secondly, it has become clear that most interior point algorithms for linear programming (LP) may be extended to semidefinite programming. The polynomial complexity of these methods and their success...
On Semidefinite Programming Relaxations of (2+p)SAT
, 2000
"... Recently, De Klerk, Van Maaren and Warners [7] investigated a relaxation of 3SAT via semidefinite programming. Thus a 3SAT formula is relaxed to a semidefinite feasibility problem. If the feasibility problem is infeasible then a certificate of unsatisfiability of the formula is obtained. The autho ..."
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Cited by 3 (0 self)
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Recently, De Klerk, Van Maaren and Warners [7] investigated a relaxation of 3SAT via semidefinite programming. Thus a 3SAT formula is relaxed to a semidefinite feasibility problem. If the feasibility problem is infeasible then a certificate of unsatisfiability of the formula is obtained. The authors proved that this approach is exact for several polynomially solvable classes of logical formulae, including 2SAT, pigeonhole formulae and mutilated chessboard formulae. In this paper we further explore this approach, and investigate the strength of the relaxation on (2 + p)SAT formulae, i.e. formulae with a fraction p of 3clauses and a fraction (1  p) of 2clauses. In the first instance, we provide an empirical computational evaluation of our approach. Secondly, we establish approximation guarantees of randomized and deterministic rounding schemes when the semidefinite feasibility problem is feasible.