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A Note on Complexity of Lp Minimization
, 2010
"... We show that the Lp (0 ≤ p < 1) minimization problem arising from sparse solution construction and compressed sensing is both hard and easy. More precisely, for any fixed 0 < p < 1, we prove that checking the global minimal value of the problem is NPHard; but computing a local minimizer of the prob ..."
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We show that the Lp (0 ≤ p < 1) minimization problem arising from sparse solution construction and compressed sensing is both hard and easy. More precisely, for any fixed 0 < p < 1, we prove that checking the global minimal value of the problem is NPHard; but computing a local minimizer of the problem is polynomialtime doable. We also develop an interiorpoint algorithm with a provable complexity bound and demonstrate preliminary computational results of effectiveness of the algorithm.
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.
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Deriving Duality for l_pnorm Optimization Using Conic Optimization
, 1999
"... In this paper, we formulate the l p norm optimization problem as a conic optimization problem and derive its standard duality properties. We first define an ad hoc closed convex cone L and derive its dual. We express then the standard l p norm optimization primal problem as a conic problem involvi ..."
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In this paper, we formulate the l p norm optimization problem as a conic optimization problem and derive its standard duality properties. We first define an ad hoc closed convex cone L and derive its dual. We express then the standard l p norm optimization primal problem as a conic problem involving L. Using convex conic duality, we derive the dual of this problem and prove the wellknown regularity properties of this primaldual pair, i.e. zero duality gap and dual attainment.
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.
Conic Optimization:
"... The purpose of this survey article is to introduce the reader to a very elegant formulation of convex optimization problems called conic optimization and outline its many advantages. ..."
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The purpose of this survey article is to introduce the reader to a very elegant formulation of convex optimization problems called conic optimization and outline its many advantages.