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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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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.
Computational Experiments With a Linear Approximation of SecondOrder Cone Optimization
, 2000
"... In this article, we present and improve a polyhedral approximation of the secondorder cone due to BenTal and Nemirovski [BTN98]. We also discuss several ways of reducing the size of this approximation. This construction allows us to approximate secondorder cone optimization problems with linear o ..."
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Cited by 4 (0 self)
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In this article, we present and improve a polyhedral approximation of the secondorder cone due to BenTal and Nemirovski [BTN98]. We also discuss several ways of reducing the size of this approximation. This construction allows us to approximate secondorder cone optimization problems with linear optimization. We implement this scheme and conduct computational experiments dealing with two classes of secondorder cone problems: the first one involves trusstopology design and uses a large number of secondorder cones with relatively small dimensions, while the second one models convex quadratic optimization problems with a single large secondorder cone.
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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Cited by 2 (0 self)
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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.
A Conic Formulation for L P Norm 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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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.
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 ...
Approximating Geometric Optimization with l_pNorm Optimization
, 2000
"... In this article, we demonstrate how to approximate geometric optimization with l p  norm optimization. These two categories of problems are well known in structured convex optimization. We describe a family of l p norm optimization problems that can be made arbitrarily close to a geometric optimiz ..."
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In this article, we demonstrate how to approximate geometric optimization with l p  norm optimization. These two categories of problems are well known in structured convex optimization. We describe a family of l p norm optimization problems that can be made arbitrarily close to a geometric optimization problem, and show that the dual problems for these approximations are also approximating the dual geometric optimization problem. Finally, we use these approximations and the duality theory for l p norm optimization to derive simple proofs of the weak and strong duality theorems for geometric optimization.
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.