A cutting plane method, using the idea of Tuy cuts, has been suggested in earlier papers as a possible means of solving reverse convex programs. However, the method is fraught with theoretical and numerical difficulti...
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A cutting plane method, using the idea of Tuy cuts, has been suggested in earlier papers as a possible means of solving reverse convex programs. However, the method is fraught with theoretical and numerical difficulties. Stringent sufficient conditions for convergence in n dimensions are given. However, examples of nonconvergence are given and reasons for this nonconvergence are developed. A result of the discussion is a convergent algorithm which combines the idea of the cutting plane method with vertex enumeration procedures in order to numerically improve upon the edge search procedure of Hillestad.
Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main t...
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Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.
Cross-manifold clustering is an extreme challenge learning problem. Since the low-density hypothesis is not satisfied in cross-manifold problems, many traditional clustering methods failed to discover the cross-manifo...
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Cross-manifold clustering is an extreme challenge learning problem. Since the low-density hypothesis is not satisfied in cross-manifold problems, many traditional clustering methods failed to discover the cross-manifold structures. In this article, we propose multiple flat projections clustering (MFPC) for cross-manifold clustering. In our MFPC, the given samples are projected into multiple localized flats to discover the global structures of implicit manifolds. Thus, the intersected clusters are distinguished in various projection flats. In MFPC, a series of nonconvex matrix optimization problems is solved by a proposed recursive algorithm. Furthermore, a nonlinear version of MFPC is extended via kernel tricks to deal with a more complex cross-manifold learning situation. The synthetic tests show that our MFPC works on the cross-manifold structures well. Moreover, experimental results on the benchmark datasets and object tracking videos show excellent performance of our MFPC compared with some state-of-the-art manifold clustering methods.
We discuss the L-p (0 <= p < 1) minimization problem arising from sparse solution construction and compressed sensing. For any fixed 0 < p < 1, we prove that finding the global minimal value of the problem...
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We discuss the L-p (0 <= p < 1) minimization problem arising from sparse solution construction and compressed sensing. For any fixed 0 < p < 1, we prove that finding the global minimal value of the problem is strongly NP-Hard, but computing a local minimizer of the problem can be done in polynomial time. We also develop an interior-point potential reduction algorithm with a provable complexity bound and demonstrate preliminary computational results of effectiveness of the algorithm.
For solving a broad class of nonconvex programming problems on an unbounded constraint set, we provide a self-adaptive step-size strategy that does not include line-search techniques and establishes the convergence of...
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For solving a broad class of nonconvex programming problems on an unbounded constraint set, we provide a self-adaptive step-size strategy that does not include line-search techniques and establishes the convergence of a generic approach under mild assumptions. Specifically, the objective function may not satisfy the convexity condition. Unlike descent line-search algorithms, it does not need a known Lipschitz constant to figure out how big the first step should be. The crucial feature of this process is the steady reduction of the step size until a certain condition is fulfilled. In particular, it can provide a new gradient projection approach to optimization problems with an unbounded constrained set. To demonstrate the effectiveness of the proposed technique for large-scale problems, we apply it to some experiments on machine learning, such as supervised feature selection, multi-variable logistic regressions and neural networks for classification.
A novel method is outlined for flexible heat exchanger network synthesis including nonconvex problems. The presented method is sequentially implemented by two main steps: structure synthesis and area optimization;neve...
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A novel method is outlined for flexible heat exchanger network synthesis including nonconvex problems. The presented method is sequentially implemented by two main steps: structure synthesis and area optimization;nevertheless, the optimization of heat exchanger areas can still react on the structure to gain the global optimal solution. The structure is initially synthesized at the nominal operating point and renewed by the topological union with the structure of the critical point and the improved heat transfer loops disconnection strategy. For area optimization, an iterative approach with strong robustness is proposed based on the influences of heat exchanger areas on flexibility index and total annual cost, respectively. The direction matrix method is employed to provide the operational flexibility of the network and the critical operating point. Two examples with nonconvex feasible regions have been studied, and the results well demonstrate the effectiveness of the proposed approach.
This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. The framework of the algorithm is taken from a pure cutting-plane decision set-ba...
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This article presents an outcome-space pure cutting-plane algorithm for globally solving the linear multiplicative programming problem. The framework of the algorithm is taken from a pure cutting-plane decision set-based method developed by Horst and Tuy for solving concave minimization problems. By adapting this method to an outcome-space reformulation of the linear multiplicative programming problem, rather than applying directly the method to the original decision-set formulation, it is expected that considerable computational savings can be obtained. Also, we show how additional computational benefits might be obtained by implementing the new algorithm appropriately. To illustrate the new algorithm, we apply it to the solution of a sample problem.
We consider the problem of designing a suboptimal H-2/H-infinity feedback control law for a linear time-invariant control system when a complete set of state variables is not available. This problem can be necessarily...
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We consider the problem of designing a suboptimal H-2/H-infinity feedback control law for a linear time-invariant control system when a complete set of state variables is not available. This problem can be necessarily restated as a nonconvex optimization problem with a bilinear, multiobjective functional under suitably chosen linear matrix inequality (LMI) constraints. To solve such a problem, we propose an LMI-based procedure which is a sequential linearization programming approach. The properties and the convergence of the algorithm are discussed in detail. Finally, several numerical examples for static H-2/H-infinity output feedback problems demonstrate the applicability of the considered algorithm and also verify the theoretical results numerically.
Optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in NASA's Hubble Space Telescope (HST). NASA's success with reconstruction algorithms on t...
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Optical wavefront reconstruction algorithms played a central role in the effort to identify gross manufacturing errors in NASA's Hubble Space Telescope (HST). NASA's success with reconstruction algorithms on the HST has led to an effort to develop software that can aid and in some cases replace complicated, expensive, and error-prone hardware. Among the many applications is HST's replacement, the Next Generation Space Telescope (NGST). This work details the theory of optical wavefront reconstruction, reviews some numerical methods for this problem, and presents a novel numerical technique that we call extended least squares. We compare the performance of these numerical methods for potential inclusion in prototype NGST optical wavefront reconstruction software. We begin with a tutorial on Rayleigh-Sommerfeld diffraction theory.
In the well-known fixed-charge linear programming problem, it is assumed, for each activity, that the value of the fixed charge incurred when the level of the activity is positive does not depend upon which other acti...
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In the well-known fixed-charge linear programming problem, it is assumed, for each activity, that the value of the fixed charge incurred when the level of the activity is positive does not depend upon which other activities, if any, are also undertaken at a positive level. However, we have encountered several practical problems where this assumption does not hold. In an earlier paper, we developed a new problem, called the interactive fixed-charge linear programming problem (IFCLP), to model these problems. In this paper, we show how to construct the convex envelopes and other convex underestimating functions for the objective function for problem (IFCLP) over various rectangular subsets of its domain. Using these results, we develop a specialized branch-and-bound algorithm for problem (IFCLP) which finds an exact optimal solution for the problem in a finite number of steps. We also discuss the main properties of this algorithm.
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