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Interiorpoint Methods
, 2000
"... The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadrati ..."
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The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by selfconcordant barrier functions.
A superlinearly convergent predictorcorrector method for degenerate LCP in a wide neighborhood of the central path with O (√n L)iteration complexity
, 2006
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Search Directions And Convergence Analysis Of Some Infeasible PathFollowing Methods For The Monotone SemiDefinite LCP
, 1996
"... We consider a family of primal/primaldual/dual search directions for the monotone LCP over the space of n \Theta n symmetric blockdiagonal matrices. We consider two infeasible predictorcorrector pathfollowing methods using these search directions, with the predictor and corrector steps used eith ..."
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Cited by 21 (2 self)
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We consider a family of primal/primaldual/dual search directions for the monotone LCP over the space of n \Theta n symmetric blockdiagonal matrices. We consider two infeasible predictorcorrector pathfollowing methods using these search directions, with the predictor and corrector steps used either in series (similar to the MizunoToddYe method) or in parallel (similar to Mizuno et al./McShane's method). The methods attain global linear convergence with a convergence ratio which, depending on the quality of the starting iterate, ranges from 1 \Gamma O( p n) \Gamma1 to 1 \Gamma O(n) \Gamma1 . Our analysis is fairly simple and parallels that for the LP and LCP cases. 1 Introduction Since the original work of Nesterov and Nemirovskii [26], followed by that of Alizadeh [1] and Jarre [11], there has been very active research on interiorpoint methods for the semidefinite linear programming problem (SDLP) and the semidefinite linear complementarity problem (SDLCP). In particular,...
Error bounds and limiting behavior of weighted paths associated with the sdp map X 1/2 SX 1/2
 SIAM JOURNAL ON OPTIMIZATION
, 2004
"... This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X1/2SX1/2 = νW, where W is a fixed positive definite matrix and ν> 0 is a parameter, under the assumption that the problem has a strict ..."
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Cited by 9 (2 self)
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This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X1/2SX1/2 = νW, where W is a fixed positive definite matrix and ν> 0 is a parameter, under the assumption that the problem has a strictly complementary primaldual optimal solution. It is shown that a weighted central path as a function of ν can be extended analytically beyond 0 and hence that the path converges as ν ↓ 0. Characterization of the limit points of the path and its normalized firstorder derivatives are also provided. As a consequence, it is shown that a weighted central path can have two types of behavior: it converges either as Θ(ν) or as Θ( ν) depending on whether the matrix W on a certain scaled space is block diagonal or not, respectively. We also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primaldual optimal solutions. Finally, in light of the results of this paper, we give a characterization of a sufficient condition proposed by Potra and Sheng which guarantees the superlinear convergence of a class of primaldual interiorpoint SDP algorithms.
Behavioral measures and their correlation with IPM iteration counts on semidefinite programming problems. USCISE working paper #200502, MIT, 2005. url: wwwrcf.usc.edu/˜fordon
"... We study four measures of problem instance behavior that might account for the observed differences in interiorpoint method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasib ..."
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Cited by 8 (1 self)
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We study four measures of problem instance behavior that might account for the observed differences in interiorpoint method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar) condition measure C(d) of the data instance, (iii) a measure of the nearabsence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the correlation between these measures and IPM iteration counts (solved using the software SDPT3) when the measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.896), and is a very good predictor of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The nearabsence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations.
On a General Class of InteriorPoint Algorithms for Semidefinite Programming with Polynomial Complexity and Superlinear Convergence
 Department of Mathematics, The University of Iowa, Iowa City, IA
, 1996
"... We propose a unified analysis for a class of infeasiblestart predictorcorrector algorithms for semidefinite programming problems, using the MonteiroZhang unified direction. The algorithms are direct generalizations of the MizunoToddYe predictorcorrector algorithm for linear programming. We show ..."
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Cited by 8 (5 self)
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We propose a unified analysis for a class of infeasiblestart predictorcorrector algorithms for semidefinite programming problems, using the MonteiroZhang unified direction. The algorithms are direct generalizations of the MizunoToddYe predictorcorrector algorithm for linear programming. We show that the algorithms belonging to this class are globally convergent, provided the problem has a solution, and have optimal computational complexity. We also give simple sufficient conditions for superlinear convergence. Our results generalize the results obtained by the first two authors for the infeasibleinteriorpoint algorithm proposed by Kojima, Shida and Shindoh and Potra and Sheng. Key Words: semidefinite programming, predictorcorrector, infeasibleinteriorpoint algorithm, polynomial complexity, superlinear convergence. Abbreviated Title: On a general class of algorithms for SDP. Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA. The work of these two ...
Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem
 Mathematical Programming, Series A
"... An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the ..."
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Cited by 4 (3 self)
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An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves offcentral paths. We study offcentral paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each offcentral path is a welldefined analytic curve with parameter µ ranging over (0,∞) and any accumulation point of the offcentral path is a solution to SDLCP. Through a simple example we show that the offcentral paths are not analytic as a function of µ and have first derivatives which are unbounded as a function of µ at µ = 0 in general. On the other hand, for the same example, we can find a subset of offcentral paths which are analytic at µ = 0. These “nice ” paths are characterized by some algebraic equations.
Solving Semidefinite Programs in Mathematica
 Dept. of Mathematics, University of Iowa
, 1996
"... Interiorpoint algorithms for solving semidefinite programs are described and implemented in Mathematica. Included are MizunoToddYe type predictorcorrector algorithms and Mehrotra type predictorcorrector algorithms. Three different search directions  the AHO direction, the KSH/HRVW/M directio ..."
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Cited by 2 (0 self)
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Interiorpoint algorithms for solving semidefinite programs are described and implemented in Mathematica. Included are MizunoToddYe type predictorcorrector algorithms and Mehrotra type predictorcorrector algorithms. Three different search directions  the AHO direction, the KSH/HRVW/M direction and the NT direction, are used. Homogeneous algorithms using the PotraSheng formulation are tested. A simple procedure is derived for the computation of the homogeneous search directions. Numerical results show that the homogeneous algorithms are generally superior over their nonhomogeneous counterparts in terms of number of iterations. Department of Computer Science, The University of Iowa, Iowa City, IA 52242, USA. y Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA. The work of the last two authors was supported in part by NSF Grant DMS 9305760. 1 Introduction In this paper we consider the semidefinite programming (SDP) problem: minfC ffl X : A i ffl X...
Superlinear Convergence of Infeasible PredictorCorrector PathFollowing Interior Point Algorithm for SDLCP using the HKM Direction
"... Interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the inter ..."
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Interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the interior of the region. In [31], these offcentral paths are shown to be welldefined analytic curves and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem (SDLCP). The study of these paths provides a way to understand how iterates generated by an interior point algorithm behave. In this paper, we give a weak sufficient condition using these offcentral paths that guarantees superlinear convergence of a predictorcorrector pathfollowing interior point algorithm for SDLCP using the HKM direction. This sufficient condition is implied by a currently known sufficient condition for superlinear convergence. Using this sufficient condition, we show that for any linear semidefinite feasibility problem, superlinear convergence using the interior point algorithm, with the HKM direction, can be achieved, for a suitable starting point. We work under the assumption of strict complementarity.
Asymptotic Behavior of Underlying NT Paths in Interior Point Method for Monotone Semidefinite Linear Complementarity Problems
"... This is the PrePublished Version. An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define th ..."
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This is the PrePublished Version. An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In [32], these offcentral paths are shown to be welldefined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In [32][34], the asymptotic behavior of offcentral paths corresponding to the HKM direction is studied. In particular, in [32], the authors study the asymptotic behavior of these paths for a simple example, while, in [33,34], the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study offcentral paths corresponding to another wellknown direction, the NesterovTodd (NT) direction. Again, we give necessary and sufficient conditions for these offcentral paths to be analytic w.r.t. µ and then w.r.t. µ, at solutions of a general SDLCP. Also, as in [32], we present offcentral path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.