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13
Proximal Point Method for a Special Class of Nonconvex Functions on Hadamard Manifolds
, 2009
"... In this paper we present the proximal point method for a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary ..."
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In this paper we present the proximal point method for a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is guaranteed. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minimizer is obtained.
A LINESEARCHBASED DERIVATIVEFREE APPROACH FOR NONSMOOTH CONSTRAINED OPTIMIZATION∗
"... Abstract. In this paper, we propose new linesearchbased methods for nonsmooth constrained optimization problems when firstorder information on the problem functions is not available. In the first part, we describe a general framework for boundconstrained problems and analyze its convergence towar ..."
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Abstract. In this paper, we propose new linesearchbased methods for nonsmooth constrained optimization problems when firstorder information on the problem functions is not available. In the first part, we describe a general framework for boundconstrained problems and analyze its convergence toward stationary points, using the Clarke–Jahn directional derivative. In the second part, we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth. In this case, we first split the constraints into two subsets: difficult general nonlinear constraints and simple bound constraints on the variables. Then, we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the boundconstrained minimization of the proposed exact penalty function. Finally, we use the framework developed for the boundconstrained case to solve the penalized problem. Moreover, we prove that every accumulation point, under standard assumptions on the search directions, of the generated sequence of iterates is a stationary point of the original constrained problem. In the last part of the paper, we report extended numerical results on both boundconstrained and nonlinearly constrained problems, showing that our approach is promising when compared to some stateoftheart codes from the literature.
A Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization *
"... Abstract In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges g ..."
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Abstract In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a ParetoClarke critical point. Our method may be seen as an extension, for the non convex case, of the inexact proximal method for multiobjective convex minimization problems studied by Bonnel et al. (SIAM Journal on Optimization 15, 4, 953970, 2005).
Sensitivity to constraints in blackbox optimization ∗
, 2010
"... Abstract: The paper proposes a framework for sensitivity analyses of blackbox constrained optimization problems for which Lagrange multipliers are not available. Two strategies are developed to analyze the sensitivity of the optimal objective function value to general constraints. These are a simple ..."
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Abstract: The paper proposes a framework for sensitivity analyses of blackbox constrained optimization problems for which Lagrange multipliers are not available. Two strategies are developed to analyze the sensitivity of the optimal objective function value to general constraints. These are a simple method which may be performed immediately after a single optimization, and a detailed method performing biobjective optimization on the minimization of the objective versus the constraint of interest. The detailed method provides points on the Pareto front of the objective versus a chosen constraint. The proposed methods are tested on an academic test case and on an engineering problem using the mesh adaptive direct search algorithm.
Benchmarking Numerical Multiobjective Optimizers Revisited
 GENETIC AND EVOLUTIONARY COMPUTATION CONFERENCE (GECCO 2015)
, 2015
"... ..."
DerivativeFree Optimization: Lifting SingleObjective to MultiObjective Algorithm
"... Abstract. Most of the derivativefree optimization (DFO) algorithms rely on a comparison function able to compare any pair of points with respect to a blackbox objective function. Recently, new dedicated derivativefree optimization algorithms have emerged to tackle multiobjective optimization pr ..."
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Abstract. Most of the derivativefree optimization (DFO) algorithms rely on a comparison function able to compare any pair of points with respect to a blackbox objective function. Recently, new dedicated derivativefree optimization algorithms have emerged to tackle multiobjective optimization problems and provide a Pareto front approximation to the user. This work aims at reusing single objective DFO algorithms (such as NelderMead) in the context of multiobjective optimization. Therefore we introduce a comparison function able to compare a pair of points in the context of a set of nondominated points. We describe an algorithm, MOGEN, which initializes a Pareto front approximation composed of a population of instances of singleobjective DFO algorithms. These algorithms use the same introduced comparison function relying on a shared Pareto front approximation. The different instances of singleobjective DFO algorithms are collaborating and competing to improve the Pareto front approximation. Our experiments comparing MOGEN with the stateof theart Direct MultiSearch algorithm on a large set of benchmarks shows the practicality of the approach, allowing to obtain high quality Pareto fronts using a reasonably small amount of function evaluations. 1
APPROACH FOR NONSMOOTH OPTIMIZATION
, 2013
"... In this paper, we propose new linesearchbased methods for nonsmooth constrained optimization problems when firstorder information on the problem functions is not available. In the first part, we describe a general framework for boundconstrained problems and analyze its convergence towards station ..."
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In this paper, we propose new linesearchbased methods for nonsmooth constrained optimization problems when firstorder information on the problem functions is not available. In the first part, we describe a general framework for boundconstrained problems and analyze its convergence towards stationary points, using the ClarkeJahn directional derivative. In the second part, we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth. In this case, we first split the constraints into two subsets: difficult general nonlinear constraints and simple bound constraints on the variables. Then, we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the boundconstrained minimization of the proposed exact penalty function. Finally, we use the framework developed for the boundconstrained case to solve the penalized problem, and we prove that every accumulation point of the generated sequence of iterates is a stationary point of the original constrained problem. In the last part of the paper, we report extended numerical results on both boundconstrained and nonlinearly
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"... Abstract. In this paper, we propose new linesearchbased methods for nonsmooth constrained optimization problems when firstorder information on the problem functions is not available. In the first part, we describe a general framework for boundconstrained problems and analyze its convergence towar ..."
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Abstract. In this paper, we propose new linesearchbased methods for nonsmooth constrained optimization problems when firstorder information on the problem functions is not available. In the first part, we describe a general framework for boundconstrained problems and analyze its convergence toward stationary points, using the Clarke–Jahn directional derivative. In the second part, we consider inequality constrained optimization problems where both objective function and constraints can possibly be nonsmooth. In this case, we first split the constraints into two subsets: difficult general nonlinear constraints and simple bound constraints on the variables. Then, we use an exact penalty function to tackle the difficult constraints and we prove that the original problem can be reformulated as the boundconstrained minimization of the proposed exact penalty function. Finally, we use the framework developed for the boundconstrained case to solve the penalized problem. Moreover, we prove that every accumulation point, under standard assumptions on the search directions, of the generated sequence of iterates is a stationary point of the original constrained problem. In the last part of the paper, we report extended numerical results on both boundconstrained and nonlinearly constrained problems, showing that our approach is promising when compared to some stateoftheart codes from the literature.
Noname manuscript No. (will be inserted by the editor) Multiobjective Traffic Engineering for Data Center Networks
"... Abstract Data centers are now the basis for many Internet and cloud computing services. The Spanning Tree Protocol and its variants have been widely used in data center networks for a couple of decades. An efficient use of the limited spanning tree links would enable to solve the traffic engineerin ..."
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Abstract Data centers are now the basis for many Internet and cloud computing services. The Spanning Tree Protocol and its variants have been widely used in data center networks for a couple of decades. An efficient use of the limited spanning tree links would enable to solve the traffic engineering problem in data centers. In this paper, we propose five local search approaches for generating a good set of spanning trees in data centers using multiple VLANs. The quality of these algorithms is evaluated by different multicriteria assessment methods. The performance of each algorithm is assessed based on three standard measures: maximal link utilization, sum load, and the number of used links.
Efficient Cardinality/MeanVariance Portfolios
"... Abstract A number of variants of the classical Markowitz meanvariance optimization model for portfolio selection have been investigated to render it more realistic. Recently, it has been studied the imposition of a cardinality constraint, setting an upper bound on the number of active positions ta ..."
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Abstract A number of variants of the classical Markowitz meanvariance optimization model for portfolio selection have been investigated to render it more realistic. Recently, it has been studied the imposition of a cardinality constraint, setting an upper bound on the number of active positions taken in the portfolio, in an attempt to improve its performance and reduce transactions costs. However, one can regard cardinality as an objective function itself, thus adding another goal to those two traditionally considered (the variance and the mean of the return). In this paper, we suggest a new approach to directly compute sparse portfolios by reformulating the cardinality constrained Markowitz meanvariance optimization model as a biobjective optimization problem, allowing the investor to analyze the efficient tradeoff between meanvariance and cardinality, in a general scenario where shortselling is allowed. Since cardinality is a nonsmooth objective function, one has chosen a derivativefree algorithm (based on direct multisearch) for the solution of the biobjective optimization problem. For the several data sets obtained from the FTSE 100 index and the Fama/French benchmark collection, direct multisearch was capable of quickly determining (insample) the efficient frontier for the biobjective cardinality/meanvariance problem. Our results showed that a number of efficient cardinality/meanvariance portfolios (with values of cardinality not high) overcome the naive strategy in terms of outofsample performance measured by the Sharpe ratio, which is known to be extremely difficult.