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Proving Strong Duality for Geometric Optimization Using a Conic Formulation
, 1999
"... Geometric optimization is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the wellknown associated duality theory, using conic optimization. After introducing suitable convex cones and studying their ..."
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Cited by 5 (1 self)
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Geometric optimization is an important class of problems that has many applications, especially in engineering design. In this article, we provide new simplified proofs for the wellknown associated duality theory, using conic optimization. After introducing suitable convex cones and studying their properties, we model geometric optimization problems with a conic formulation, which allows us to apply the powerful duality theory of conic optimization and derive the duality results valid for geometric optimization.
Improving Complexity of Structured Convex Optimization Problems Using SelfConcordant Barriers
, 2001
"... The purpose of this paper is to provide improved complexity results for several classes of structured convex optimization problems using to the theory of selfconcordant functions developed in [11]. We describe the classical shortstep interiorpoint method and optimize its parameters in order to pr ..."
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The purpose of this paper is to provide improved complexity results for several classes of structured convex optimization problems using to the theory of selfconcordant functions developed in [11]. We describe the classical shortstep interiorpoint method and optimize its parameters in order to provide the best possible iteration bound. We also discuss the necessity of introducing two parameters in the definition of selfconcordancy and which one is the best to fix. A lemma from [3] is improved, which allows us to review several classes of structured convex optimization problems and improve the corresponding complexity results.
Topics In Convex Optimization: InteriorPoint Methods, Conic Duality and Approximations
"... Contents Table of Contents i List of gures v Preface vii Introduction 1 I INTERIORPOINT METHODS 5 1 Interiorpoint methods for linear optimization 7 1.1.1 Linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 The simplex method . . . . . . . . . . . . . . . . . . . . ..."
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Contents Table of Contents i List of gures v Preface vii Introduction 1 I INTERIORPOINT METHODS 5 1 Interiorpoint methods for linear optimization 7 1.1.1 Linear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.2 The simplex method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 A rst glimpse on interiorpoint methods . . . . . . . . . . . . . . . . 9 1.1.4 A short historical account . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Building blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Newton's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.4 Barrier function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.5 The central path . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An Extended Conic Formulation for Geometric Optimization
, 2000
"... The author has recently proposed a new way of formulating two classical classes of structured convex problems, geometric and l p norm optimization, using dedicated convex cones [Gli99, GT00]. This approach has some advantages over the traditional formulation: it simplifies the proofs of the wellkn ..."
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The author has recently proposed a new way of formulating two classical classes of structured convex problems, geometric and l p norm optimization, using dedicated convex cones [Gli99, GT00]. This approach has some advantages over the traditional formulation: it simplifies the proofs of the wellknown associated duality properties (i.e. weak and strong duality) and the design of a polynomial algorithm becomes straightforward. These new proofs rely on the general duality theory valid for convex problems expressed in conic form [SW70, Stu00] and the work on polynomial interiorpoint methods by Nesterov and Nemirovsky [NN94]. In this paper, we make a step towards the description of a common framework that would include these two classes of problems. Indeed, we introduce an extended variant of the cone for geometric optimization used in [Gli99] and show it is equally suitable to formulate this class of problems. This new cone has the additional advantage of being very similar to the ...
SelfConcordant Functions in Structured Convex Optimization
, 2000
"... This paper provides a selfcontained introduction to the theory of selfconcordant functions [8] and applies it to several classes of structured convex optimization problems. We describe the classical shortstep interiorpoint method and optimize its parameters to provide its best possible iteration ..."
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This paper provides a selfcontained introduction to the theory of selfconcordant functions [8] and applies it to several classes of structured convex optimization problems. We describe the classical shortstep interiorpoint method and optimize its parameters to provide its best possible iteration bound. We also discuss the necessity of introducing two parameters in the definition of selfconcordancy, how they react to addition and scaling and which one is the best to fix. A lemma from [2] is improved and allows us to review several classes of structured convex optimization problems and evaluate their algorithmic complexity, using the selfconcordancy of the associated logarithmic barriers.
ON THE LASSERRE HIERARCHY OF SEMIDEFINITE PROGRAMMING RELAXATIONS OF CONVEX POLYNOMIAL OPTIMIZATION PROBLEMS
"... Abstract. The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995–2014, 2009.] We give a ..."
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Abstract. The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995–2014, 2009.] We give a new proof of the finite convergence property, that does not require the assumption that the Hessian of the objective be positive definite on the entire feasible set, but only at the optimal solution. In addition, we show that the number of steps needed for convergence depends on more than the input size of the problem. In particular, the size of the semidefinite program that gives the exact reformulation of the convex polynomial optimization problem may be exponential in the input size. Key words. convex polynomial optimization, sum of squares of polynomials, positivstellensatz, semidefinite programming. AMS subject classifications. 90C60, 90C56, 90C26 1. Polynomial optimization and the Lasserre hierarchy. We consider the polynomial optimization problem pmin =min