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17
Convex Nondifferentiable Optimization: A Survey Focussed On The Analytic Center Cutting Plane Method.
, 1999
"... We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct pr ..."
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Cited by 60 (2 self)
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We present a survey of nondifferentiable optimization problems and methods with special focus on the analytic center cutting plane method. We propose a selfcontained convergence analysis, that uses the formalism of the theory of selfconcordant functions, but for the main results, we give direct proofs based on the properties of the logarithmic function. We also provide an in depth analysis of two extensions that are very relevant to practical problems: the case of multiple cuts and the case of deep cuts. We further examine extensions to problems including feasible sets partially described by an explicit barrier function, and to the case of nonlinear cuts. Finally, we review several implementation issues and discuss some applications.
A Cutting Plane Method from Analytic Centers for Stochastic Programming
 Mathematical Programming
, 1994
"... The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decompositi ..."
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Cited by 52 (18 self)
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The stochastic linear programming problem with recourse has a dual block angular structure. It can thus be handled by Benders decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block angular structure and can be handled by DantzigWolfe decomposition the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization. 1 Introduction The study of optimization problems in the presence of uncertainty still taxes the limits of methodology and software. One of the most approachable settings is that of twostaged planning under uncertainty, in which a first stage decision has to be taken bef...
PrimalDual TargetFollowing Algorithms for Linear Programming
 ANNALS OF OPERATIONS RESEARCH
, 1993
"... In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Al ..."
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Cited by 26 (1 self)
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In this paper we propose a method for linear programming with the property that, starting from an initial noncentral point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Along with the convergence analysis we provide a general framework which enables us to analyze various primaldual algorithms in the literature in a short and uniform way.
Distributionally Robust Optimization under Moment Uncertainty with Application to DataDriven Problems
"... Stochastic programs can effectively describe the decisionmaking problem in an uncertain environment. Unfortunately, such programs are often computationally demanding to solve. In addition, their solutions can be misleading when there is ambiguity in the choice of a distribution for the random param ..."
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Cited by 23 (2 self)
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Stochastic programs can effectively describe the decisionmaking problem in an uncertain environment. Unfortunately, such programs are often computationally demanding to solve. In addition, their solutions can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model describing one’s uncertainty in both the distribution’s form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance). We demonstrate that for a wide range of cost functions the associated distributionally robust stochastic program can be solved efficiently. Furthermore, by deriving new confidence regions for the mean and covariance of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. This is confirmed in a practical example of portfolio selection, where our framework leads to better performing policies on the “true” distribution underlying the daily return of assets.
On the Comparative Behavior of Kelley's Cutting Plane Method and the Analytic Center Cutting plane Method
, 1996
"... In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method applied to convex optimization problems, which may lead to weaker results than Kelley's cutting plane method. Improvements to the analytic center cutting plane method are suggested. 1 I ..."
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Cited by 12 (8 self)
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In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method applied to convex optimization problems, which may lead to weaker results than Kelley's cutting plane method. Improvements to the analytic center cutting plane method are suggested. 1 Introduction In this paper, we explore a weakness of a specific implementation of the analytic center cutting plane method, and propose improvements. Cutting plane algorithms are designed to solve general convex optimization problems. They assume that the only information available around the current iterate takes the form of cutting planes, either supporting hyperplanes to the epigraph of the objective function, or separating hyperplanes from the feasible set. The two types of hyperplanes jointly define a linear programming, polyhedral, relaxation of the original convex optimization problem. The key issue in designing a specific cutting plane algorithm is the choice of a point in the current poly...
A Generic PathFollowing Algorithm With a Sliding Constraint and Its Application to Linear Programming and the Computation of Analytic Centers
, 1996
"... We propose a generic pathfollowing scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic pr ..."
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Cited by 8 (5 self)
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We propose a generic pathfollowing scheme which is essentially a method of centers that can be implemented with a variety of algorithms. The complexity estimate is computed on the sole assumption that a certain local quadratic convergence property holds, independently of the specific algorithmic procedure in use, primal, dual or primaldual. We show convergence in O( p n) iterations. We verify that the primal, dual and primaldual algorithms satisfy the local quadratic convergence property. The method can be applied to solve the linear programming problem (with a feasible start) and to compute the analytic center of a bounded polytope. The generic pathfollowing scheme easily extends to the logarithmic penalty barrier approach. Keywords Interior point method, method of centers, pathfollowing, linear programming. This work has been completed with the support from the Swiss National Foundation for Scientific Research, grant 1234002.92. 1 Introduction Shortly after Karmarkar's s...
J.P.: Interior point methods for nondifferentiable optimization. In: Kischka, P. et al (eds
 Operations Research Proceedings
, 1997
"... ..."
A TwoCut Approach in the Analytic Center Cutting Plane Method
, 1998
"... We analyze the two cut generation scheme in the analytic center cutting plane method. We propose an optimal updating direction when the two cuts are central. The direction is optimal in the sense that it maximizes the product of the new slacks within the trust region defined by Dikin's ellipsoi ..."
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Cited by 4 (2 self)
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We analyze the two cut generation scheme in the analytic center cutting plane method. We propose an optimal updating direction when the two cuts are central. The direction is optimal in the sense that it maximizes the product of the new slacks within the trust region defined by Dikin's ellipsoid. We prove convergence in O ( n 2 " 2 ) calls to the oracle and that the recovery of a new analytic center can be done in O(1) primal damped Newton steps. Keywords Primal Newton algorithm, Analytic center, Cutting Plane Method, Two cuts. This work has been completed with the support from the Fonds National Suisse de la Recherche Scientifique, grant 1242503.94, from the Natural Sciences and Engineering Research Council of Canada, grant number OPG0004152 and from the FCAR of Quebec. GERAD/Faculty of Management, McGill University, 1001, Sherbrooke West, Montreal, Que., H3A 1G5, Canada. Email: ma56@musica.mcgill.ca. LOGILAB/Management Studies, University of Geneva, 102, Bd CarlVogt,...
Sensitivity Analysis And The Analytic Central Path
, 1998
"... The analytic central path for linear programming has been studied because of its desirable convergence properties. This dissertation presents a detailed study of the analytic central path under perturbation of both the righthand side and cost vectors for a linear program. The analysis is divided int ..."
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Cited by 2 (1 self)
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The analytic central path for linear programming has been studied because of its desirable convergence properties. This dissertation presents a detailed study of the analytic central path under perturbation of both the righthand side and cost vectors for a linear program. The analysis is divided into three parts: extensions of results required by the convergence analysis when the data is unperturbed to include that case of data perturbation, marginal analysis of the analytic center solution with respect to linear changes in the righthand side, and parametric analysis of the analytic central path under simultaneous changes in both the righthand side and cost vectors. To extend the established convergence results when the data is fixed, it is rst shown that the union of the elements comprising a portion of the perturbed analytic central paths is bounded. This guarantees the existence of subsequences that converge, but these subsequences are not guaranteed to have the same limit without further restrictions on the data movement. Sufficient conditions are provided to insure that the limit is the analytic center of the iii limiting polytope. Furthermore, as long at the data converges and the parameter of the path is approaching zero, certain components of the the analytic central path are forced to zero. Since the introduction of the analytic center to the mathematical programming community, the analytic central path has been known to be analytic in both the righthand side and cost vectors. However, since the objective function is a continuous, piecewise linear function of the righthand side, the analytic center solution is not differentiable. We show that this solution is continuous and is infinitely, continuously, onesided differentiable. Furthermore, the analytic center sol...
Shallow, Deep and Very Deep Cuts in the Analytic Center Cutting Plane Method
, 1996
"... The analytic center cutting plane (ACCPM) methods aims to solve nondifferentiable convex problems. The technique consists of building an increasingly refined polyhedral approximation of the solution set. The linear inequalities that define the approximation are generated by an oracle as hyperplanes ..."
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Cited by 2 (0 self)
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The analytic center cutting plane (ACCPM) methods aims to solve nondifferentiable convex problems. The technique consists of building an increasingly refined polyhedral approximation of the solution set. The linear inequalities that define the approximation are generated by an oracle as hyperplanes separating a query point from the solution set. In ACCPM, the query point is the analytic center, or an approximation of it, for the current polyhedral relaxation. A primal projective algorithm is used to recover feasibility and then centrality. In this paper we show that the cut does not need to go through the query point: it can be deep or shallow. The primal framework leads to a simple analysis of the potential variation, which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step. Keywords Projective algorithm, Analytic center, cutting plane method. This work has been completed with the support from the Fonds N...