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Epigraphical cones I
 JOURNAL OF CONVEX ANALYSIS, 2011, IN PRESS
"... Up to orthogonal transformation, a solid closed convex cone K in the Euclidean space Rn+1 is the epigraph of a nonnegative sublinear function f: Rn → R. This work explores the link between the geometric properties of K and the analytic properties of f. ..."
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Up to orthogonal transformation, a solid closed convex cone K in the Euclidean space Rn+1 is the epigraph of a nonnegative sublinear function f: Rn → R. This work explores the link between the geometric properties of K and the analytic properties of f.
An interiorpoint method for the singlefacility location problem with mixed norms using a conic formulation
"... Abstract We consider the singlefacility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in R n , where each distance can be measured according to a different pnorm. We show how this problem can be expressed into a st ..."
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Abstract We consider the singlefacility location problem with mixed norms, i.e. the problem of minimizing the sum of the distances from a point to a set of fixed points in R n , where each distance can be measured according to a different pnorm. We show how this problem can be expressed into a structured conic format by decomposing the nonlinear components of the objective into a series of constraints involving threedimensional cones. Using the availability of a selfconcordant barrier for these cones, we present a polynomialtime algorithm (a longstep pathfollowing interiorpoint scheme) to solve the problem up to a given accuracy. Finally, we report computational results for this algorithm and compare with standard nonlinear optimization solvers applied to this problem.
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.