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25
Graph Cut based Inference with Cooccurrence Statistics
"... Abstract. Markov and Conditional random fields (CRFs) used in computer vision typically model only local interactions between variables, as this is computationally tractable. In this paper we consider a class of global potentials defined over all variables in the CRF. We show how they can be readily ..."
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Abstract. Markov and Conditional random fields (CRFs) used in computer vision typically model only local interactions between variables, as this is computationally tractable. In this paper we consider a class of global potentials defined over all variables in the CRF. We show how they can be readily optimised using standard graph cut algorithms at little extra expense compared to a standard pairwise field. This result can be directly used for the problem of class based image segmentation which has seen increasing recent interest within computer vision. Here the aim is to assign a label to each pixel of a given image from a set of possible object classes. Typically these methods use random fields to model local interactions between pixels or superpixels. One of the cues that helps recognition is global object cooccurrence statistics, a measure of which classes (such as chair or motorbike) are likely to occur in the same image together. There have been several approaches proposed to exploit this property, but all of them suffer from different limitations and typically carry a high computational cost, preventing their application on large images. We find that the new model we propose produces an improvement in the labelling compared to just using a pairwise model. 1
A New unblocking technique to warmstart interior point methods based on sensitivity analysis
, 2007
"... One of the main drawbacks associated with Interior Point Methods (IPM) is the perceived lack of an efficient warmstarting scheme which would enable the use of information from a previous solution of a similar problem. Recently there has been renewed interest in the subject. A common problem with war ..."
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One of the main drawbacks associated with Interior Point Methods (IPM) is the perceived lack of an efficient warmstarting scheme which would enable the use of information from a previous solution of a similar problem. Recently there has been renewed interest in the subject. A common problem with warmstarting for IPM is that an advanced starting point which is close to the boundary of the feasible region, as is typical, might lead to blocking of the search direction. Several techniques have been proposed to address this issue. Most of these aim to lead the iterate back into the interior of the feasible region we classify them as either “modification steps” or “unblocking steps ” depending on whether the modification is taking place before solving the modified problem to prevent future problems, or during the solution if and when problems become apparent. A new “unblocking” strategy is suggested which attempts to directly address the issue of blocking by performing sensitivity analysis on the Newton step with the aim of increasing the size of the step that can be taken. This analysis is used in a new technique to warmstart
Finding a point in the relative interior of a polyhedron
, 2007
"... A new initialization or ‘Phase I ’ strategy for feasible interior point methods for linear programming is proposed that computes a point on the primaldual central path associated with the linear program. Provided there exist primaldual strictly feasible points — an allpervasive assumption in inte ..."
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Cited by 6 (3 self)
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A new initialization or ‘Phase I ’ strategy for feasible interior point methods for linear programming is proposed that computes a point on the primaldual central path associated with the linear program. Provided there exist primaldual strictly feasible points — an allpervasive assumption in interior point method theory that implies the existence of the central path — our initial method (Algorithm 1) is globally Qlinearly and asymptotically Qquadratically convergent, with a provable worstcase iteration complexity bound. When this assumption is not met, the numerical behaviour of Algorithm 1 is highly disappointing, even when the problem is primaldual feasible. This is due to the presence of implicit equalities, inequality constraints that hold as equalities at all the feasible points. Controlled perturbations of the inequality constraints of the primaldual problems are introduced — geometrically equivalent to enlarging the primaldual feasible region and then systematically contracting it back to its initial shape — in order for the perturbed problems to satisfy the assumption. Thus Algorithm 1 can successfully be employed to solve each of the perturbed problems. We show that, when there exist primaldual strictly feasible points of the original problems, the resulting method, Algorithm 2, finds such a point in a finite number of changes to the perturbation parameters. When implicit equalities are present, but the original problem and its dual are feasible, Algorithm 2 asymptotically detects all the primaldual implicit equalities and generates a point in the relative interior of the primaldual feasible set. Algorithm 2 can also asymptotically detect primaldual infeasibility. Successful numerical experience with Algorithm 2 on linear programs from NETLIB and CUTEr, both with and without any significant preprocessing of the problems, indicates that Algorithm 2 may be used as an algorithmic preprocessor for removing implicit equalities, with theoretical guarantees of convergence. 1
Infeasible ConstraintReduced InteriorPoint Methods for Linear Optimization ∗
, 2010
"... Constraintreduction schemes have been proposed for the solution by interiorpoint methods of linear programs with many more inequality constraints than variables in standard dual form. Such schemes have been shown to be provably convergent and highly efficient in practice. A critical requirement of ..."
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Cited by 6 (4 self)
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Constraintreduction schemes have been proposed for the solution by interiorpoint methods of linear programs with many more inequality constraints than variables in standard dual form. Such schemes have been shown to be provably convergent and highly efficient in practice. A critical requirement of these schemes is the availability of an initial dualfeasible point. In this paper, building on a general framework (which encompasses several previously proposed approaches) for dualfeasible constraintreduced interiorpoint optimization, for which we prove convergence to a single point of the sequence of dual iterates, we propose a framework for “infeasible ” constraintreduced interiorpoint optimization. Central to this framework is an exact (ℓ1 or ℓ∞) penalty function scheme endowed with a mechanism for iterative adjustment of the penalty parameter, which aims at yielding, after a finite number of iterations, a value that guarantees feasibility (for the original problem) of the minimizers. Finiteness of the sequence of penalty parameter adjustments is proved under mild assumptions for all algorithms that fit within the framework, including “infeasible ” extensions of a “dual ” algorithm proposed in the early 1990s and of two recently proposed “primaldual ” algorithms. One of the latter two, a constraintreduced variant of Mehrotra’s PredictorCorrector algorithm, is then more specifically considered: further convergence results are proved, and numerical results are reported that demonstrate that the approach is of practical interest.
On InteriorPoint Warmstarts for Linear and Combinatorial Optimization
, 2008
"... Despite the many advantages of interiorpoint algorithms over activeset methods for linear optimization, one of the remaining practical challenges is their current limitation to efficiently solve series of related problems by an effective warmstarting strategy. In its remedy, in this paper we prese ..."
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Cited by 5 (0 self)
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Despite the many advantages of interiorpoint algorithms over activeset methods for linear optimization, one of the remaining practical challenges is their current limitation to efficiently solve series of related problems by an effective warmstarting strategy. In its remedy, in this paper we present a new infeasibleinteriorpoint approach to quickly reoptimize an initial problem instance after data perturbations, or a new linear programming relaxation after adding cutting planes for discrete or combinatorial problems. Based on the detailed complexity analysis of the underlying algorithm, we perform a comparative analysis to coldstart initialization schemes and present encouraging computational results with iteration savings around 50 % on average for perturbations of the Netlib linear programs and successive LP relaxations of maxcut and the travelingsalesman problem.
Warmstarting the Homogeneous and SelfDual Interior Point Method for Linear and Conic Quadratic Problems
 MATHEMATICAL PROGRAMMING COMPUTATION
"... We present two strategies for warmstarting primaldual interior point methods for the homogeneous selfdual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and their negli ..."
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Cited by 3 (1 self)
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We present two strategies for warmstarting primaldual interior point methods for the homogeneous selfdual model when applied to mixed linear and quadratic conic optimization problems. Common to both strategies is their use of only the final (optimal) iterate of the initial problem and their negligible computational cost. This is a major advantage when compared to previously suggested strategies that require a pool of iterates from the solution process of the initial problem. Consequently our strategies are better suited for users who use optimization algorithms as blackbox routines which usually only output the final solution. Our two strategies differ in that one assumes knowledge only of the final primal solution while the other assumes the availability of both primal and dual solutions. We analyze the strategies and deduce conditions under which they result in improved theoretical worstcase complexity. We present extensive computational results showing work reductions when warmstarting compared to coldstarting in the range 30%–75 % depending on the problem class and magnitude of the problem perturbation. The computational experiments thus substantiate that the warmstarting strategies are useful in practice.
CONVERGENCE ANALYSIS OF AN INTERIORPOINT METHOD FOR NONCONVEX NONLINEAR PROGRAMMING
"... Abstract. In this paper, we present global and local convergence results for an interiorpoint method for nonlinear programming. The algorithm uses an ℓ1 penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using ..."
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Abstract. In this paper, we present global and local convergence results for an interiorpoint method for nonlinear programming. The algorithm uses an ℓ1 penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interiorpoint method [6]. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Qquadratically. The proposed approach improves on existing results in several ways: (1) the convergence analysis does not assume boundedness of dual iterates, (2) local convergence does not require the Linear Independence Constraint Qualification, (3) the solution of the penalty problem is shown to locally converge to optima that may not satisfy the KarushKuhnTucker conditions, and (4) the algorithm is applicable to mathematical programs with equilibrium constraints. 1.
Review of mixedinteger nonlinear and generalized disjunctive programming applications in Process Systems Engineering
, 2014
"... In this chapter we present some of the applications of MINLP and generalized disjunctive programming (GDP) in process systems engineering (PSE). For a comprehensive review of mixedinteger nonlinear optimization we refer the reader to the work by Belotti et al.[1]. Bonami et al.[2] review convex MI ..."
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In this chapter we present some of the applications of MINLP and generalized disjunctive programming (GDP) in process systems engineering (PSE). For a comprehensive review of mixedinteger nonlinear optimization we refer the reader to the work by Belotti et al.[1]. Bonami et al.[2] review convex MINLP algorithms and software in more detail. Tawarmalani and Sahinidis[3] describe global optimization theory,
Using interiorpoint methods within an outer approximation framework for mixedinteger nonlinear programming
 IMAMINLP Issue
"... Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via o ..."
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Abstract. Interiorpoint methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixedinteger nonlinear programming problems via outer approximation. However, traditionally, infeasible primaldual interiorpoint methods have had two main perceived deficiencies: (1) lack of infeasibility detection capabilities, and (2) poor performance after a warmstart. In this paper, we propose the exact primaldual penalty approach as a means to overcome these deficiencies. The generality of this approach to handle any change to the problem makes it suitable for the outer approximation framework, where each nonlinear subproblem can differ from the others in the sequence in a variety of ways. Additionally, we examine cases where the nonlinear subproblems take on special forms, namely those of secondorder cone programming problems and semidefinite programming problems. Encouraging numerical results are provided.