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32
Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization
 SIAM Journal on Optimization
, 1993
"... We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to S ..."
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Cited by 482 (11 self)
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We study the semidefinite programming problem (SDP), i.e the problem of optimization of a linear function of a symmetric matrix subject to linear equality constraints and the additional condition that the matrix be positive semidefinite. First we review the classical cone duality as specialized to SDP. Next we present an interior point algorithm which converges to the optimal solution in polynomial time. The approach is a direct extension of Ye's projective method for linear programming. We also argue that most known interior point methods for linear programs can be transformed in a mechanical way to algorithms for SDP with proofs of convergence and polynomial time complexity also carrying over in a similar fashion. Finally we study the significance of these results in a variety of combinatorial optimization problems including the general 01 integer programs, the maximum clique and maximum stable set problems in perfect graphs, the maximum k partite subgraph problem in graphs, and va...
Interior Point Methods For Optimal Control Of DiscreteTime Systems
 Journal of Optimization Theory and Applications
, 1993
"... . We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete ..."
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Cited by 31 (5 self)
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. We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete time linearquadratic regulator problem with mixed state/control constraints, and show how it can be efficiently incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interiorpoint method is the narrowbanded structure of the coefficient matrix which is factorized at each iteration. Key words. interior point algorithms, optimal control, banded linear systems. 1. Introduction. The problem of optimal control of an initial value ordinary differential equation, with Bolza objectives and mixed constraints, is min x;u Z T 0 L(x(t); u(t); t) dt + OE f (x(T )); x(t) = f(x(t); u(t); t); x(0) = x init ; (1.1) g(x(t); u(...
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 15 (8 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
Degeneracy in Interior Point Methods for Linear Programming
, 1991
"... ... In this paper, we survey the various theoretical and practical issues related to degeneracy in IPM's for linear programming. We survey results which for the most part already appeared in the literature. Roughly speaking, we shall deal with four topics: the effect of degeneracy on the convergence ..."
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Cited by 11 (1 self)
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... In this paper, we survey the various theoretical and practical issues related to degeneracy in IPM's for linear programming. We survey results which for the most part already appeared in the literature. Roughly speaking, we shall deal with four topics: the effect of degeneracy on the convergence of IPM's, on the trajectories followed by the algorithms, the effect of degeneracy in numerical performance, and on finding basic solutions.
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 11 (5 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.
New Complexity Analysis of the PrimalDual Newton Method for Linear Optimization
, 1998
"... We deal with the primaldual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the pra ..."
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Cited by 11 (7 self)
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We deal with the primaldual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the practical behavior of the algorithms and the theoretical performance results, in favor of the practical behavior. This is especially true for socalled largeupdate methods. We present some new analysis tools, based on a proximity measure introduced by Jansen et al., in 1994, that may help to close this gap. This proximity measure has not been used in the analysis of largeupdate method before. Our new analysis not only provides a unified way for the analysis of both largeupdate and smallupdate methods, but also improves the known iteration bounds.
A New PrimalDual InteriorPoint Method for Semidefinite Programming
, 1994
"... Semidefinite programming (SDP) is a convex optimization problem in the space of symmetric matrices. Primaldual interiorpoint methods for SDP are discussed. These generate primal and dual matrices X and Z which commute only in the limit. A new method is proposed which iterates in the space of commu ..."
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Cited by 10 (1 self)
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Semidefinite programming (SDP) is a convex optimization problem in the space of symmetric matrices. Primaldual interiorpoint methods for SDP are discussed. These generate primal and dual matrices X and Z which commute only in the limit. A new method is proposed which iterates in the space of commuting matrices. Let S! n\Thetan denote the set of real symmetric n \Theta n matrices. The standard inner product on this space is A ffl B = tr AB = P i;j a ij b ij : By X 0, where X 2 S! n\Thetan , we mean that X is positive semidefinite. Consider the semidefinite programming problem (SDP) min C ffl X (1) s:t: A i ffl X = b i i = 1; . . . ; m; X 0: (2) Here C and A i , i = 1; . . . ; m, are all fixed matrices in S! n\Thetan , and the unknown variable X also lies in S! n\Thetan . The semidefinite constraint on X is said to be nonsmooth, since it is equivalent to a bound constraint on the least eigenvalue of X , which is not a differentiable function of X . The constraint is, how...
Theoretical Convergence of LargeStep PrimalDual Interior Point Algorithms for Linear Programming
 Mathematical Programming
, 1992
"... . This paper proposes two sets of rules, Rule G and Rule P, for controlling step lengths in a generic primaldual interior point method for solving the linear programming problem in standard form and its dual. Theoretically, Rule G ensures the global convergence, while Rule P, which is a special cas ..."
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Cited by 8 (0 self)
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. This paper proposes two sets of rules, Rule G and Rule P, for controlling step lengths in a generic primaldual interior point method for solving the linear programming problem in standard form and its dual. Theoretically, Rule G ensures the global convergence, while Rule P, which is a special case of Rule G, ensures the O(nL) iteration polynomialtime computational complexity. Both rules depend only on the lengths of the steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions. They rely neither on neighborhoods of the central trajectory nor on potential function. These rules allow large steps without performing any line search. Rule G is especially flexible enough for implementation in practically efficient primaldual interior point algorithms. Key words: PrimalDual Interior Point Algorithm, Linear Program, Large Step, Global Convergence, PolynomialTime Convergence Abbreviated Title: LargeStep PrimalDual...
On Effectively Computing the Analytic Center of the Solution Set By PrimalDual InteriorPoint Methods
, 1997
"... The computation of the analytic center of the solution set can be important in linear programming applications where it is desirable to obtain a solution that is not near the relative boundary of the solution set. In this work we discuss the effective computation of the analytic center solution by t ..."
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Cited by 7 (4 self)
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The computation of the analytic center of the solution set can be important in linear programming applications where it is desirable to obtain a solution that is not near the relative boundary of the solution set. In this work we discuss the effective computation of the analytic center solution by the use of primaldual interiorpoint methods. A primaldual interiorpoint algorithm designed for effectively computing the analyticcenter solution is proposed and numerical results are presented. This research was partially supported by NSF Cooperative Agreement No. CCR88 09615, NSF Grant DMS 9305760, ARO Grant 9DAAL0390G0093 DOE Grant DEFG0586ER25017, and AFOSR Grant 890363. y Department of Computational and Applied Mathematics and Center for Research on Parallel Computation, Rice University, Houston, TX 772511892. Partially supported by Fulbright/LASPAU. z Department of Computational and Applied Mathematics and Center for Research on Parallel Computation, Rice University,...
On InteriorPoint Newton Algorithms For Discretized Optimal Control Problems With State Constraints
 OPTIM. METHODS SOFTW
, 1998
"... In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive ..."
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Cited by 7 (2 self)
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In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive an affinescaling and two primaldual interiorpoint Newton algorithms by applying, in an interiorpoint way, Newton's method to equivalent forms of the firstorder optimality conditions. Under appropriate assumptions, the interiorpoint Newton algorithms are shown to be locally welldefined with a qquadratic rate of local convergence. By using the structure of the problem, the linear algebra of these algorithms can be reduced to the null space of the Jacobian of the equality constraints. The similarities between the three algorithms are pointed out, and their corresponding versions for the general nonlinear programming problem are discussed.