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Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones
, 1998
"... SeDuMi is an addon for MATLAB, that lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This pape ..."
Abstract

Cited by 736 (3 self)
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SeDuMi is an addon for MATLAB, that lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox.
On the closedness of the linear image of a closed convex cone
, 1992
"... informs doi 10.1287/moor.1060.0242 ..."
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 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
On sensitivity of central solutions in semidefinite programming
 Math. Program
, 1998
"... In this paper we study the properties of the analytic central path of a semide nite programming problem under perturbation of a set of input parameters. Speci cally, we analyze the behavior of solutions on the central path with respect to changes on the right hand side of the constraints, including ..."
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Cited by 8 (2 self)
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In this paper we study the properties of the analytic central path of a semide nite programming problem under perturbation of a set of input parameters. Speci cally, we analyze the behavior of solutions on the central path with respect to changes on the right hand side of the constraints, including the limiting behavior when the central optimal solution is approached. Our results are of interest for the sake ofnumerical analysis, sensitivity analysis and parametric programming. Under the primaldual Slater condition and the strict complementarity condition we show that the derivatives of central solutions with respect to the right hand side parameters converge as the path tends to the central optimal solution. Moreover, the derivatives are bounded, i.e. a Lipschitz constant exists. This Lipschitz constant can be thought of as a condition number for the semide nite programming problem. It is a generalization of the familiar condition number for linear equation systems and linear programming problems. However, the generalized condition number depends on the right hand side parameters as well, whereas it is wellknown that in the linear programming case the condition number depends only on the constraint matrix. We demonstrate that the existence of strictly complementary solutions is important for the Lipschitz constant to exist. Moreover, we give an example in which the set of right hand side parameters for which the strict complementarity condition holds is neither open nor closed. This is remarkable since a similar set for which the primaldual Slater condition holds is always open. Key words: analytic central path, semide nite programming, sensitivity, condition number.
Error Bounds for Linear Matrix Inequalities
, 1998
"... For iterative sequences that converge to the solution set of a linear matrix inequality, we show that the distance of the iterates to the solution set is at most O(ffl 2 \Gammad ). The nonnegative integer d is the socalled degree of singularity of the linear matrix inequality, and ffl denotes th ..."
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Cited by 4 (0 self)
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For iterative sequences that converge to the solution set of a linear matrix inequality, we show that the distance of the iterates to the solution set is at most O(ffl 2 \Gammad ). The nonnegative integer d is the socalled degree of singularity of the linear matrix inequality, and ffl denotes the amount of constraint violation in the iterate. For infeasible linear matrix inequalities, we show that the minimal norm of fflapproximate primal solutions is at least 1=O(ffl 1=(2 d \Gamma1) ), and the minimal norm of fflapproximate Farkas type dual solutions is at most O(1=ffl 2 d \Gamma1 ). As an application of these error bounds, we show that for any bounded sequence of fflapproximate solutions to a semidefinite programming problem, the distance to the optimal solution set is at most O(ffl 2 \Gammak ), where k is the degree of singularity of the optimal solution set. Keywords: semidefinite programming, error bounds, linear matrix inequality, regularized duality. AMS s...
A Simple Derivation of a Facial Reduction Algorithm and Extended Dual Systems
"... The Facial Reduction Algorithm (FRA) of Borwein and Wolkowicz and the Extended Dual System (EDS) of Ramana aim to better understand duality, when a conic linear system Ax K b (P) has no strictly feasible solution. We ffl provide a simple proof of the correctness of a variant of FRA. ffl show how ..."
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Cited by 4 (0 self)
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The Facial Reduction Algorithm (FRA) of Borwein and Wolkowicz and the Extended Dual System (EDS) of Ramana aim to better understand duality, when a conic linear system Ax K b (P) has no strictly feasible solution. We ffl provide a simple proof of the correctness of a variant of FRA. ffl show how it naturally leads to the validity of a family of extended dual systems. ffl Summarize, which subsets of K related to the system (P) (as the minimal cone and its dual) have an extended representation. 1 Introduction Farkas' lemma assuming a CQ Duality results for the conic linear system Ax K b (P) 1 A Facial Reduction Algorithm and Extended Dual Systems 2 are usually derived assuming some constraint qualification (CQ). The most frequently used CQ is strict feasibility, ie. assuming the existence of a x with A x ! K b. Here K is a closed convex cone, A : X ! Y a linear operator, with X and Y being euclidean spaces. We write z K y, and z ! K y to mean that y  z is in K, or in ri K...
Global Error Bounds for Convex Conic Problems
, 1998
"... In this paper Lipschitzian type error bounds are derived for general convex conic problems under various regularity conditions. Specifically, it is shown that if the recession directions satisfy Slater's condition then a global Lipschitzian type error bound holds. Alternatively, if the feasible regi ..."
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Cited by 2 (1 self)
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In this paper Lipschitzian type error bounds are derived for general convex conic problems under various regularity conditions. Specifically, it is shown that if the recession directions satisfy Slater's condition then a global Lipschitzian type error bound holds. Alternatively, if the feasible region is bounded, then the ordinary Slater condition guarantees a global Lipschitzian type error bound. These can be considered as generalizations of previously known results for inequality systems. Moreover, some of the results are also generalized to the intersection of multiple cones. Under Slater's condition alone, a global Lipschitzian type error bound may not hold. However, it is shown that such an error bound holds for a specific region. For linear systems we show that the constant involved in Hoffman's error bound can be estimated by the socalled condition number for linear programming. Key words: Error bound, convex conic problems, LMIs, condition number. AMS subject classification: 5...