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31
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.
Multiple Cuts in the Analytic Center Cutting Plane Method
, 1998
"... We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables wi ..."
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Cited by 31 (1 self)
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We analyze the multiple cut generation scheme in the analytic center cutting plane method. We propose an optimal primal and dual updating direction when the cuts are central. The direction is optimal in the sense that it maximizes the product of the new dual slacks and of the new primal variables within the trust regions defined by Dikin's primal and dual ellipsoids. The new primal and dual directions use the variancecovariance matrix of the normals to the new cuts in the metric given by Dikin's ellipsoid. We prove that the recovery of a new analytic center from the optimal restoration direction can be done in O(p log(p + 1)) damped Newton steps, where p is the number of new cuts added by the oracle, which may vary with the iteration. The results and the proofs are independent of the specific scaling matrix primal, dual or primaldual that is used in the computations. The computation of the optimal direction uses Newton's method applied to a selfconcordant function of p variab...
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.
Towards a Practical Volumetric Cutting Plane Method for Convex Programming
 SIAM Journal on Optimization
, 1997
"... We consider the volumetric cutting plane method for finding a point in a convex set C ae ! n that is characterized by a separation oracle. We prove polynomiality of the algorithm with each added cut placed directly through the current point, and show that this "central cut" version of th ..."
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Cited by 19 (2 self)
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We consider the volumetric cutting plane method for finding a point in a convex set C ae ! n that is characterized by a separation oracle. We prove polynomiality of the algorithm with each added cut placed directly through the current point, and show that this "central cut" version of the method can be implemented using no more than 25n constraints at any time. Currently visiting the Center for Operations Research and Econometrics, Catholic University of Louvain, LouvainlaNeuve, Belgium, with support from a CORE fellowship. 1 Introduction Let C ae ! n be a convex set. Given a point ¯ x 2 ! n , a separation oracle for C either reports that ¯ x 2 C, or returns a separating hyperplane a 2 ! n such that a T x ? a T ¯ x for every x 2 C. The convex feasibility problem is to use such an oracle to find a point in C, or prove that the volume of C must be less than that of an ndimensional sphere of radius 2 \GammaL , for given L ? 0. It is well known [9] that a variety of...
The Analytic Center Cutting Plane Method with Semidefinite Cuts
 SIAM JOURNAL ON OPTIMIZATION
, 2000
"... We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution s ..."
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Cited by 18 (1 self)
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We analyze an analytic center cutting plane algorithm for the convex feasibility problems with semidefinite cuts. At each iteration the oracle returns a pdimensional semidefinite cut at an approximate analytic center of the set of localization. The set of localization, which contains the solution set, is a compact set consists of piecewise algebraic surfaces. We prove that the analytic center is recovered after adding a pdimensional cut in O(p log(p 1)) damped Newton's iteration. We also prove that the algorithm stops when the dimension of the accumulated block diagonal matrix cut reaches to the bound of O (p 2 m 3 =ffl 2 ), where p is the maximum dimension cut and ffl is radius of the largest ball contained in the solution set.
A LogBarrier Method With Benders Decomposition For Solving TwoStage Stochastic Programs
 Mathematical Programming 90
, 1999
"... An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving twostage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite numb ..."
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Cited by 17 (6 self)
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An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving twostage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run in polynomialtime. Key Words: Stochastic programming, Largescale linear programming, Barrier function, Interior point methods, Benders decomposition, Complexity. Abbreviated Title: A logbarrier method with Benders decomposition AMS(MOS) subject classifications: 90C15, 90C05, 90C06, 90C60. 1 1. Introduction In this paper we propose an algorithm for solving twostage stochastic programs, establish fundamental properties of the algorithm, and analyze the convergence. An example of a twostage stochastic program is a production planning problem. The production and demand take place in the first and second periods, resp...
INTERIOR POINT METHODS FOR COMBINATORIAL OPTIMIZATION
, 1995
"... Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivale ..."
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Cited by 16 (9 self)
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Research on using interior point algorithms to solve combinatorial optimization and integer programming problems is surveyed. This paper discusses branch and cut methods for integer programming problems, a potential reduction method based on transforming an integer programming problem to an equivalent nonconvex quadratic programming problem, interior point methods for solving network flow problems, and methods for solving multicommodity flow problems, including an interior point column generation algorithm.
A long step cutting plane algorithm that uses the volumetric barrier
, 1995
"... A cutting plane method for linear/convex programming is described. It is based on the volumetric barrier, introduced by Vaidya. The algorithm is a long step one, and has a complexity of O(n1.5L) Newton steps. This is better than the O(n √ mL) complexity of noncutting plane long step methods based o ..."
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Cited by 8 (5 self)
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A cutting plane method for linear/convex programming is described. It is based on the volumetric barrier, introduced by Vaidya. The algorithm is a long step one, and has a complexity of O(n1.5L) Newton steps. This is better than the O(n √ mL) complexity of noncutting plane long step methods based on the volumetric barrier, but it is however worse than Vaidya’s original O(nL) result (which is not a long step algorithm). Major features of our algorithm are that when adding cuts we add them right through the current point, and when seeking progress in the objective, the duality gap is reduced by half (not provably true for Vaidya’s original algorithm). Further, we generate primal as well as dual iterates, making this applicable in the column generation context as well. Vaidya’s algorithm has been used as a subroutine to obtain the best complexity for several combinatorial optimization problems – e.g, the HeldKarp lower bound for the Traveling Salesperson Problem. While our complexity result is weaker, this long step cutting plane algorithm is likely to be computationally more promising on such combinatorial optimization problems with an exponential number of constraints. We also discuss a multiple cuts version — where upto p ≤ n ‘selectively orthonormalized ’ cuts are added through the current point. This has a complexity of O(n1.5Lp log p) quasi Newton steps.
An interior point cutting plane method for the convex feasibility problem with secondorder cone inequalities
, 2004
"... ..."
A Nonlinear Analytic Center Cutting Plane Method For A Class Of Convex Programming Problems
 SIAM Journal on Optimization
, 1996
"... . A cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented. The algorithm is based on approximate analytic centers. The nonlinearity of the objective function is taken into account, yet the feasible region is approximated by a cont ..."
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Cited by 6 (2 self)
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. A cutting plane algorithm for minimizing a convex function subject to constraints defined by a separation oracle is presented. The algorithm is based on approximate analytic centers. The nonlinearity of the objective function is taken into account, yet the feasible region is approximated by a containing polytope. This containing polytope is regularly updated by adding a new cut through a test point. Each test point is an approximate analytic center of the intersection of a containing polytope and a level set of the nonlinear objective function. We establish the complexity of the algorithm. Our complexity estimate is given in terms of the problem dimension, the desired accuracy of an approximate solution and other parameters that depend on the geometry of a specific instance of the problem. Key words. convex programming, interiorpoint methods, analytic center, cutting planes, potential function, selfconcordance AMS subject classifications. 90C06, 90C25 1. Introduction. Recently, ...