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Register a new userOn unconstrained optimization problems solved using CDT and triality theory
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C. Zalinescu
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DY Gao solely or together with some of his collaborators applied his Canonical duality theory (CDT) for solving a class of unconstrained optimization problems, getting the so-called triality theorems. Unfortunately, the double-min duality from these results published before 2010 revealed to be false, even if in 2003 DY Gao announced that certain additional conditions are needed for getting it. After 2010 DY Gao together with some of his collaborators published several papers in which they added additional conditions for getting double-min and double-max dualities in the triality theorems. The aim of this paper is to treat rigorously this kind of problems and to discuss several results concerning the triality theory obtained up to now.
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The Fujitsu Digital Annealer (DA) is designed to solve fully connected quadratic unconstrained binary optimization (QUBO) problems. It is implemented on application-specific CMOS hardware and currently solves problems of up to 1024 variables. The DAs algorithm is currently based on simulated annealing; however, it differs from it in its utilization of an efficient parallel-trial scheme and a dynamic escape mechanism. In addition, the DA exploits the massive parallelization that custom application-specific CMOS hardware allows. We compare the performance of the DA to simulated annealing and parallel tempering with isoenergetic cluster moves on two-dimensional and fully connected spin-glass problems with bimodal and Gaussian couplings. These represent the respective limits of sparse versus dense problems, as well as high-degeneracy versus low-degeneracy problems. Our results show that the DA currently exhibits a time-to-solution speedup of roughly two orders of magnitude for fully connected spin-glass problems with bimodal or Gaussian couplings, over the single-core implementations of simulated annealing and parallel tempering Monte Carlo used in this study. The DA does not appear to exhibit a speedup for sparse two-dimensional spin-glass problems, which we explain on theoretical grounds. We also benchmarked an early implementation of the Parallel Tempering DA. Our results suggest an improved scaling over the other algorithms for fully connected problems of average difficulty with bimodal disorder. The next generation of the DA is expected to be able to solve fully connected problems up to 8192 variables in size. This would enable the study of fundamental physics problems and industrial applications that were previously inaccessible using standard computing hardware or special-purpose quantum annealing machines.
In this paper we focus on the unconstrained binary quadratic optimization model, maximize x^t Qx, x binary, and consider the problem of identifying optimal solutions that are robust with respect to perturbations in the Q matrix.. We are motivated to find robust, or stable, solutions because of the uncertainty inherent in the big data origins of Q and limitations in computer numerical precision, particularly in a new class of quantum annealing computers. Experimental design techniques are used to generate a diverse subset of possible scenarios, from which robust solutions are identified. An illustrative example with practical application to business decision making is examined. The approach presented also generates a surface response equation which is used to estimate upper bounds in constant time for Q instantiations within the scenario extremes. In addition, a theoretical framework for the robustness of individual x_i variables is considered by examining the range of Q values over which the x_i are predetermined.
We present a gradient-based algorithm for unconstrained minimization derived from iterated linear change of basis. The new method is equivalent to linear conjugate gradient in the case of a quadratic objective function. In the case of exact line search it is a secant method. In practice, it performs comparably to BFGS and DFP and is sometimes more robust.
Quadratic Unconstrained Binary Optimization models are useful for solving a diverse range of optimization problems. Constraints can be added by incorporating quadratic penalty terms into the objective, often with the introduction of slack variables needed for conversion of inequalities. This transformation can lead to a significant increase in the size and density of the problem. Herein, we propose an efficient approach for recasting inequality constraints that reduces the number of linear and quadratic variables. Experimental results illustrate the efficacy.
We provide a condition-based analysis of two interior-point methods for unconstrained geometric programs, a class of convex programs that arise naturally in applications including matrix scaling, matrix balancing, and entropy maximization. Our condition numbers are natural geometric quantities associated with the Newton polytope of the geometric program, and lead to diameter bounds on approximate minimizers. We also provide effective bounds on the condition numbers both in general and under combinatorial assumptions on the Newton polytope. In this way, we generalize the iteration complexity of recent interior-point methods for matrix scaling and matrix balancing. Recently, there has been much work on algorithms for certain optimization problems on Lie groups, known as capacity and scaling problems. For commutative groups, these problems reduce to unconstrained geometric programs, which serves as a particular source of motivation for our work.
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