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49
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 ..."
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Cited by 1334 (4 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.
Solving Euclidean Distance Matrix Completion Problems Via Semidefinite Programming
, 1997
"... Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by ..."
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Cited by 88 (15 self)
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Given a partial symmetric matrix A with only certain elements specified, the Euclidean distance matrix completion problem (IgDMCP) is to find the unspecified elements of A that make A a Euclidean distance matrix (IgDM). In this paper, we follow the successful approach in [20] and solve the IgDMCP by generalizing the completion problem to allow for approximate completions. In particular, we introduce a primaldual interiorpoint algorithm that solves an equivalent (quadratic objective function) semidefinite programming problem (SDP). Numerical results are included which illustrate the efficiency and robustness of our approach. Our randomly generated problems consistently resulted in low dimensional solutions when no completion existed.
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 73 (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
Using SeDuMi 1.0x , A Matlab TOOLBOX FOR OPTIMIZATION OVER SYMMETRIC CONES
, 1999
"... SeDuMi is an addon for MATLAB, which 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 p ..."
Abstract

Cited by 46 (0 self)
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SeDuMi is an addon for MATLAB, which 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.
Implementation of interior point methods for mixed semidefinite and second order cone optimization problems
 Optimization Methods and Software
"... There is a large number of implementational choices to be made for the primaldual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different resear ..."
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Cited by 40 (0 self)
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There is a large number of implementational choices to be made for the primaldual interior point method in the context of mixed semidefinite and second order cone optimization. This paper presents such implementational issues in a unified framework, and compares the choices made by different research groups. This is also the first paper to provide an elaborate discussion of the implementation in SeDuMi.
Optimal design of IIR digital filters with robust stability using conicquadraticprogramming updates
 IEEE Trans. Signal Processing
, 2003
"... In this paper, minimax design of infiniteimpulseresponse (IIR) filters with prescribed stability margin is formulated as a conic quadratic programming (CQP) problem. CQP is known as a class of wellstructured convex programming problems for which efficient interiorpoint solvers are available. By c ..."
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Cited by 15 (2 self)
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In this paper, minimax design of infiniteimpulseresponse (IIR) filters with prescribed stability margin is formulated as a conic quadratic programming (CQP) problem. CQP is known as a class of wellstructured convex programming problems for which efficient interiorpoint solvers are available. By considering factorized denominators, the proposed formulation incorporates a set of linear constraints that are sufficient and near necessary for the IIR filter to have a prescribed stability margin. Also included in the formulation is a secondorder cone condition on the magnitude of each update that ensures the validity of a key linear approximation used in the design and eliminates a linesearch step. Collectively, these features lead to improved designs relative to several established methods. 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 semidefinite programming problem under perturbation of a set of input parameters. Specifically, we analyze the behavior of solutions on the central path with respect to changes on the right hand side of the constraints, includin ..."
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Cited by 11 (2 self)
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In this paper we study the properties of the analytic central path of a semidefinite programming problem under perturbation of a set of input parameters. Specifically, 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 of numerical 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 semidefinite 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.
Pattern Separation Via Ellipsoids and Conic Programming
, 1998
"... this document. The first chapter is about mathematical programming. We will start by describing how and why researchers were led to study special types of mathematical programs, namely convex programs and conic programs. We will also provide a detailed discussion about conic duality and give a class ..."
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Cited by 11 (0 self)
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this document. The first chapter is about mathematical programming. We will start by describing how and why researchers were led to study special types of mathematical programs, namely convex programs and conic programs. We will also provide a detailed discussion about conic duality and give a classification of conic programs. We will then describe what are selfscaled cones and why they are so useful in conic programming. Finally, we will give an overview of what can be modelled using a SQL conic program, keeping in mind our pattern separation problem. Since most of the material in the chapter is standard, many of the proofs are omitted. The second chapter will concentrate on pattern separation. After a short description of the problem, we will successively describe four different separation methods using SQL conic programming. For each method, various properties are investigated. Each algorithm has in fact been successively designed with the objective of eliminating the drawbacks of the previous one, CONTENTS 3 while keeping its good properties. We conclude this chapter with a small section describing the state of the art in pattern separation with ellipsoids. The third chapter reports some computational experiments with our four methods, and provides a comparison with other separation procedures. Finally, we conclude this work by providing a short summary, highlighting the author's personal contribution and giving some interesting perspectives for further research. Chapter 1 Conic programming 1.1 Introduction
A Conic Formulation for l_pNorm Optimization
, 2000
"... In this paper, we formulate the l p norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone L p , study its properties and derive its dual. This allows us to express ..."
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Cited by 10 (1 self)
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In this paper, we formulate the l p norm optimization problem as a conic optimization problem, derive its standard duality properties and show it can be solved in polynomial time. We first define an ad hoc closed convex cone L p , study its properties and derive its dual. This allows us to express the standard l p norm optimization primal problem as a conic problem involving L p . Using convex conic duality and our knowledge about L p , we proceed to derive the dual of this problem and prove the wellknown regularity properties of this primaldual pair, i.e. zero duality gap and primal attainment. Finally, we prove that the class of l p norm optimization problems can be solved up to a given accuracy in polynomial time, using the framework of interiorpoint algorithms and selfconcordant barriers.