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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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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.
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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Cited by 1 (0 self)
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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 ...
Impact Factor: 1.852
"... In recent years cone constraint optimization problems has been favored by scholars, especially for the secondorder cone constraints programming problem,they have carried on the detailed study, had got the corresponding theoretical achievements. In this paper, on the basis of the existing cone const ..."
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In recent years cone constraint optimization problems has been favored by scholars, especially for the secondorder cone constraints programming problem,they have carried on the detailed study, had got the corresponding theoretical achievements. In this paper, on the basis of the existing cone constraint optimization problem, we puts forward the definition of projection secondorder cone,we related properties of this cone and the corresponding convexity function, the linear projected secondorder cone constraint programme problem, the dual problem, the optimality conditions,we can converted it into corresponding secondorder cone programme problems.
ON IMPLEMENTATION OF A SELFDUAL EMBEDDING METHOD FOR CONVEX PROGRAMMING∗
, 2004
"... In this paper, we implement Zhang’s method [22], which transforms a general convex optimization problem with smooth convex constraints into a convex conic optimization problem and then apply the techniques of selfdual embedding and central path following for solving the resulting conic optimizatio ..."
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In this paper, we implement Zhang’s method [22], which transforms a general convex optimization problem with smooth convex constraints into a convex conic optimization problem and then apply the techniques of selfdual embedding and central path following for solving the resulting conic optimization model. A crucial advantage of the approach is that no initial solution is required, and the method is particularly suitable when the feasibility status of the problem is unknown. In our implementation, we use a merit function approach proposed by Andersen and Ye [1] to determine the step size along the search direction. We evaluate the efficiency of the proposed algorithm by observing its performance on some test problems, which include logarithmic functions, exponential functions and quadratic functions in the constraints. Furthermore, we consider in particular the geometric programming and Lpprogramming problems. Numerical results of our algorithm on these classes of optimization problems are reported. We conclude that the algorithm is stable, efficient and easytouse in general. As the method allows the user to freely select the initial solution if he/she so wishes, it is natural to take advantage of this and apply the socalled warmstart strategy, whenever the data of a new problem is not too much different from a previously solved problem. This strategy turns out to be effective, according to our numerical experience.
Bad semidefinite programs: they all look the same
"... Duality theory plays a central role in semidefinite programming, since in optimization algorithms a dual solution serves as a certificate of optimality. However, in semidefinite duality pathological phenomena occur: nonattainment of the optimal value, positive duality gaps, and infeasibility of the ..."
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Duality theory plays a central role in semidefinite programming, since in optimization algorithms a dual solution serves as a certificate of optimality. However, in semidefinite duality pathological phenomena occur: nonattainment of the optimal value, positive duality gaps, and infeasibility of the dual, even when the primal is bounded. We say that the semidefinite system PSD = {x  ∑ m i=1 xiAi ≼ B} is badly behaved, or lacks uniform LPduality, if for some linear objective function c the value sup{〈c,x〉x ∈ PSD} is finite, but the dual program has no solution attaining the same value. We give simple, and exact characterizations of badly behaved semidefinite systems. Surprisingly, it turns out that the system α 1 1 0 x1 ≼ , (0.1)
with mixed norms using a
, 2007
"... An interiorpoint method for the singlefacility location problem with mixed norms using a conic formulation ..."
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An interiorpoint method for the singlefacility location problem with mixed norms using a conic formulation
Conic Optimization: An Elegant Framework for Convex Optimization
, 2001
"... 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. After a brief introduction to convex optimization, the notion of convex cone is introduced, which leads to the conic ..."
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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. After a brief introduction to convex optimization, the notion of convex cone is introduced, which leads to the conic formulation of convex optimization problems. This formulation features a very symmetric dual problem, and several useful duality theorems pertaining to this conic primaldual pair are presented. The usefulness of this approach is then demonstrated with its application to a wellknown class of convex problems called l p norm optimization. A suitably defined convex cone leads to a conic formulation for this problem, which allows us to derive its dual and the associated weak and strong duality properties in a seamless manner.
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.