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37
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.
Cuts for mixed 01 conic programming
, 2005
"... In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of ti ..."
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Cited by 25 (0 self)
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In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 01 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 01 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.
A Cutting Plane Algorithm for Large Scale Semidefinite Relaxations
 In The sharpest cut, MPS/SIAM Ser. Optim
, 2001
"... The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 01 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with re ..."
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Cited by 20 (3 self)
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The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 01 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges to the optimal solution under reasonable assumptions on the separation oracle and the feasible set. We have implemented a practical variant of the cutting plane algorithm for improving semidefinite relaxations of constrained quadratic 01 programming problems by oddcycle inequalities. We also consider separating oddcycle inequalities with respect to a larger support than given by the cost matrix and present a heuristic for selecting this support. Our preliminary computational results for maxcut instances on toroidal grid graphs and balanced bisection instances indicate that warm start is highly efficient and that enlarging the support may sometimes improve the quality of relaxations considerably.
Polynomial interior point cutting plane methods
 Optimization Methods and Software
, 2003
"... Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approxim ..."
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Cited by 20 (9 self)
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Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the relaxation. Typically, these cutting plane methods can be developed so as to exhibit polynomial convergence. The volumetric cutting plane algorithm achieves the theoretical minimum number of calls to a separation oracle. Longstep versions of the algorithms for solving convex optimization problems are presented. 1
Optimizing call center staffing using simulation and analytic center cutting plane methods
 Management Science
, 2005
"... We consider the problem of minimizing staffing costs in an inbound call center, while maintaining an acceptable level of service in multiple time periods. The problem is complicated by the fact that staffing level in one time period can affect the service levels in subsequent periods. Moreover, sta ..."
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Cited by 19 (0 self)
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We consider the problem of minimizing staffing costs in an inbound call center, while maintaining an acceptable level of service in multiple time periods. The problem is complicated by the fact that staffing level in one time period can affect the service levels in subsequent periods. Moreover, staff schedules typically take the form of shifts covering several periods. Interactions between staffing levels in different time periods, as well as the impact of shift requirements on the staffing levels and cost should be considered in the planning. Traditional staffing methods based on stationary queueing formulas do not take this into account. We present a simulationbased analytic center cutting plane method to solve a sample average approximation of the problem. We establish convergence of the method when the service level functions are discrete pseudoconcave. An extensive numerical study of a moderately large call center shows that the method is robust and, in most of the test cases, outperforms traditional staffing heuristics that are based on analytical queueing methods.
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.
Cutting plane methods for semidefinite programming
, 2002
"... A semidefinite programming problem is a nonsmooth optimization problem, so it can be solved using a cutting plane approach. In this paper, we analyze properties of such an algorithm. We discuss characteristics of good polyhedral representations for the semidefinite program. We show that the complexi ..."
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Cited by 13 (7 self)
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A semidefinite programming problem is a nonsmooth optimization problem, so it can be solved using a cutting plane approach. In this paper, we analyze properties of such an algorithm. We discuss characteristics of good polyhedral representations for the semidefinite program. We show that the complexity of an interior point cutting plane approach based on a semiinfinite formulation of the semidefinite program has complexity comparable with that of a direct interior point solver. We show that cutting planes can always be found efficiently that support the feasible region. Further, we characterize the supporting hyperplanes that give high dimensional tangent planes, and show how such supporting hyperplanes can be found efficiently.
A semidefinite programming based polyhedral cut and price approach for the maxcut problem
 Computational Optimization and Applications
, 2006
"... Abstract. We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the wellknown SDP relaxation of maxcut is formulated as a semiinfinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; ..."
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Cited by 13 (5 self)
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Abstract. We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the wellknown SDP relaxation of maxcut is formulated as a semiinfinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this constitutes the pricing (column generation) phase of the algorithm. Cutting planes based on the polyhedral theory of the maxcut problem are then added to the primal problem in order to improve the SDP relaxation; this is the cutting phase of the algorithm. We provide computational results, and compare these results with a standard SDP cutting plane scheme. semidefinite programming, column generation, cutting plane methods, combinatorial optimizaKeywords: tion