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13
Segmentation of multivariate mixed data via lossy coding and compression
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2007
"... Abstract—In this paper, based on ideas from lossy data coding and compression, we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions, which are allowed to be almost degenerate. The goal is to find the optimal segmen ..."
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Cited by 68 (13 self)
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Abstract—In this paper, based on ideas from lossy data coding and compression, we present a simple but effective technique for segmenting multivariate mixed data that are drawn from a mixture of Gaussian distributions, which are allowed to be almost degenerate. The goal is to find the optimal segmentation that minimizes the overall coding length of the segmented data, subject to a given distortion. By analyzing the coding length/rate of mixed data, we formally establish some strong connections of data segmentation to many fundamental concepts in lossy data compression and ratedistortion theory. We show that a deterministic segmentation is approximately the (asymptotically) optimal solution for compressing mixed data. We propose a very simple and effective algorithm that depends on a single parameter, the allowable distortion. At any given distortion, the algorithm automatically determines the corresponding number and dimension of the groups and does not involve any parameter estimation. Simulation results reveal intriguing phasetransitionlike behaviors of the number of segments when changing the level of distortion or the amount of outliers. Finally, we demonstrate how this technique can be readily applied to segment real imagery and bioinformatic data. Index Terms—Multivariate mixed data, data segmentation, data clustering, rate distortion, lossy coding, lossy compression, image segmentation, microarray data clustering. 1
Canonical dual transformation method and generalized triality theory in nonsmooth global optimization
 Journal of Global Optimization
"... Abstract. This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in R n c ..."
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Cited by 17 (9 self)
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Abstract. This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in R n can be reformulated into certain smooth/convex unconstrained dual problems in R m with m � n and without duality gap, and some NPhard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.
Convex envelopes of multilinear functions over a unit hypercube and over special discrete sets. Acta mathematica vietnamica
"... Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. In this paper, we present some general as well as explicit characterizations of the convex envelope of multilinear functions defined over a unit hypercube. A new approach is used to derive this characterization via a related ..."
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Cited by 15 (1 self)
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Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. In this paper, we present some general as well as explicit characterizations of the convex envelope of multilinear functions defined over a unit hypercube. A new approach is used to derive this characterization via a related convex hull representation obtained by applying the ReformulationLinearization Technique (RLT) of Sherali and Adams (1990, 1994). For the special cases of multilinear functions having coefficients that are either all +1 or all −1, we develop explicit formulae for the corresponding convex envelopes. Extensions of these results are given for the case when the multilinear function is defined over discrete sets, including explicit formulae for the foregoing special cases when this discrete set is represented by generalized upper bounding (GUB) constraints in binary variables. For more general cases of multilinear functions, we also discuss how this construct can be used to generate suitable relaxations for solving nonconvex optimization problems that include such structures. 1.
Efficient exact pvalue computation for small sample, sparse, and surprising categorical data
 J. of Comp. Bio
, 2004
"... A major obstacle in applying various hypothesis testing procedures to datasets in bioinformatics is the computation of ensuing pvalues. In this paper, we define a generic branchandbound approach to efficient exact pvalue computation and enumerate the required conditions for successful application ..."
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Cited by 9 (1 self)
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A major obstacle in applying various hypothesis testing procedures to datasets in bioinformatics is the computation of ensuing pvalues. In this paper, we define a generic branchandbound approach to efficient exact pvalue computation and enumerate the required conditions for successful application. Explicit procedures are developed for the entire Cressie–Read family of statistics, which includes the widely used Pearson and likelihood ratio statistics in a oneway frequency table goodnessoffit test. This new formulation constitutes a first practical exact improvement over the exhaustive enumeration performed by existing statistical software. The general techniques we develop to exploit the convexity of many statistics are also shown to carry over to contingency table tests, suggesting that they are readily extendible to other tests and test statistics of interest. Our empirical results demonstrate a speedup of orders of magnitude over the exhaustive computation, significantly extending the practical range for performing exact tests. We also show that the relative speedup gain increases as the null hypothesis becomes sparser, that computation precision increases with increase in speedup, and that computation time is very moderately affected by the magnitude of the computed pvalue. These qualities make our algorithm especially appealing in the regimes of small samples, sparse null distributions, and rare events, compared to the alternative asymptotic approximations and Monte Carlo samplers. We discuss several established bioinformatics applications, where small sample size, small expected counts in one or more categories (sparseness), and very small pvalues do occur. Our computational framework could be applied in these, and similar cases, to improve performance. Key words: pvalue, exact tests, branch and bound, real extension, categorical data.
Reformulation and Convex Relaxation Techniques for Global Optimization
 4OR
, 2004
"... Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested i ..."
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Cited by 9 (7 self)
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Many engineering optimization problems can be formulated as nonconvex nonlinear programming problems (NLPs) involving a nonlinear objective function subject to nonlinear constraints. Such problems may exhibit more than one locally optimal point. However, one is often solely or primarily interested in determining the globally optimal point. This thesis is concerned with techniques for establishing such global optima using spatial BranchandBound (sBB) algorithms.
NONCONVEX SEMILINEAR PROBLEMS AND CANONICAL DUALITY SOLUTIONS
"... This paper presents a brief review and some new developments on the canonical duality theory with applications to a class of variational problems in nonconvex mechanics and global optimization. These nonconvex problems are directly related to a large class of semilinear partial differential equatio ..."
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Cited by 8 (7 self)
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This paper presents a brief review and some new developments on the canonical duality theory with applications to a class of variational problems in nonconvex mechanics and global optimization. These nonconvex problems are directly related to a large class of semilinear partial differential equations in mathematical physics including phase transitions, postbuckling of large deformed beam model, chaotic dynamics, nonlinear field theory, and superconductivity. Numerical discretizations of these equations lead to a class of very difficult global minimization problems in finite dimensional space. It is shown that by the use of the canonical dual transformation, these nonconvex constrained primal problems can be converted into certain very simple canonical dual problems. The criticality condition leads to dual algebraic equations which can be solved completely. Therefore, a complete set of solutions to these very difficult primal problems can be obtained. The extremality of these solutions are controlled by the socalled triality theory. Several examples are illustrated including the nonconvex constrained quadratic programming. Results show that these very difficult primal problems can be converted into certain simple canonical (either convex or concave) dual problems, which can be solved completely. Also some very interesting new phenomena, i.e. triochaos and metachaos, are discovered in postbuckling of nonconvex systems. The author believes that these important phenomena exist in many nonconvex dynamical systems and deserve to have a detailed study.
Sufficient conditions and perfect duality in nonconvex minimization with inequality constraints
 J. Industrial and Management Optimization
"... Abstract. This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. ..."
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Cited by 5 (3 self)
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Abstract. This paper presents a duality theory for solving concave minimization problem and nonconvex quadratic programming problem subjected to nonlinear inequality constraints. By use of the canonical dual transformation developed recently, two canonical dual problems are formulated, respectively. These two dual problems are perfectly dual to the primal problems with zero duality gap. It is proved that the sufficient conditions for global minimizers and local extrema (both minima and maxima) are controlled by the triality theory discovered recently [5]. This triality theory can be used to develop certain useful primaldual methods for solving difficult nonconvex minimization problems. Results shown that the difficult quadratic minimization problem with quadratic constraint can be converted into a onedimensional dual problem, which can be solved completely to obtain all KKT points and global minimizer. 1. Concave Minimization Problem and Parametrization. The concave minimization problem to be discussed in this paper is denoted as the primal problem ((P) in short)
Fractional programming with convex quadratic forms and functions
 European Journal of Operational Research
"... This article is concerned with two global optimization problems (P1) and (P2). Each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function, where at least one of the convex functions is a quadratic form. First, the articl ..."
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Cited by 2 (0 self)
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This article is concerned with two global optimization problems (P1) and (P2). Each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function, where at least one of the convex functions is a quadratic form. First, the article presents and validates a number of theoretical properties of these problems. Included among these properties is the result that, under a mild assumption, any globally optimal solution for problem (P1) must belong to the boundary of its feasible region. Also among these properties is a result that shows that problem (P2) can be reformulated as a convex maximization problem. Second, the article presents for the first time an algorithm for globally solving problem (P2). The algorithm is a branch and bound algorithm in which the main computational effort involves solving a sequence of convex programming problems. Convergence properties of the algorithm are presented, and computational issues that arise in implementing the algorithm are discussed. Preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem (P2), provided that the number of variables is not too large.
On Tuy's 1964 Cone Splitting Algorithm for Concave Minimization
, 1997
"... Since the work of Zwart, it is known that cycling may occur in the cone splitting algorithm proposed by Tuy in 1964 to minimize a concave function over a polytope. In this paper, we show that despite this fact, Tuy's algorithm is convergent in the sense that it always finds an optimal solution. This ..."
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Cited by 2 (1 self)
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Since the work of Zwart, it is known that cycling may occur in the cone splitting algorithm proposed by Tuy in 1964 to minimize a concave function over a polytope. In this paper, we show that despite this fact, Tuy's algorithm is convergent in the sense that it always finds an optimal solution. This is also true for a variant of Tuy's algorithm proposed by Gallo, in which a cone is split into a smaller subset of subcones (in term of inclusion). We show on an example that this variant may also cycle. The transformation of both algorithms into finite ones is discussed.
Probability Density Function Estimation using the MinMax Measure
, 2000
"... The problem of initial probability assignment consistent with the available information about a probabilistic system is called a direct problem. Jaynes' maximum entropy principle (MaxEnt) provides a method for solving direct problems when the available information is in the form of moment constraint ..."
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Cited by 2 (0 self)
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The problem of initial probability assignment consistent with the available information about a probabilistic system is called a direct problem. Jaynes' maximum entropy principle (MaxEnt) provides a method for solving direct problems when the available information is in the form of moment constraints. On the other hand, given a probability distribution, the problem of finding a set of constraints which makes the given distribution a maximum entropy distribution is called an inverse problem. A method based on the MinMax measure to solve the above inverse problem is presented here. The MinMax measure of information, defined by Kapur, Baciu and Kesavan [1], is a quantitative measure of the information contained in a given set of moment constraints. It is based on both maximum and minimum entropy. Computational issues in the determination of the MinMax measure arising from the complexity in arriving at minimum entropy probability distributions (MinEPD) are discussed. The method to solve i...