This paper presents a linear decomposition approach for a class of nonconvex programming problems by dividing the input space into polynomially many grids. It shows that under certain assumptions the original problem ...
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This paper presents a linear decomposition approach for a class of nonconvex programming problems by dividing the input space into polynomially many grids. It shows that under certain assumptions the original problem can be transformed and decomposed into a polynomial number of equivalent linear programming subproblems. Based on solving a series of liner programming subproblems corresponding to those grid points we can obtain the near-optimal solution of the original problem. Compared to existing results in the literature, the proposed algorithm does not require the assumptions of quasi-concavity and differentiability of the objective function, and it differs significantly giving an interesting approach to solving the problem with a reduced running time.
This paper presents a robust canonical duality-triality theory for solving nonconvex programming problems under data uncertainty. This theory includes a robust canonical saddle-point theorem and robust canonical optim...
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This paper presents a robust canonical duality-triality theory for solving nonconvex programming problems under data uncertainty. This theory includes a robust canonical saddle-point theorem and robust canonical optimality conditions, which can be used to identify both robust global and local extrema of the primal problem. Two numerical examples are presented to illustrate that the robust Triality theory is particularly powerful for solving nonconvex optimization problems with data uncertainty.
In this paper, for finding a minimal efficient solution of nonconvex multiobjective programming, a constraint shifting combined homotopy is constructed and the global convergence is obtained under some mild conditions...
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In this paper, for finding a minimal efficient solution of nonconvex multiobjective programming, a constraint shifting combined homotopy is constructed and the global convergence is obtained under some mild conditions. The method requires that the initial point needs to be only in the shifted feasible set not necessarily in the original feasible set, and the normal cone condition need only be satisfied in the boundary of the shifted feasible set not the original constraint set. Some numerical tests are done and made comparison with a combined homotopy interior point method. The results show that our method is feasible and effective.
In this paper, we consider nonconvex differentiable programming under linear and nonlinear differentiable constraints. A reduced gradient and GRG (generalized reduced gradient) descent methods involving stochastic per...
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In this paper, we consider nonconvex differentiable programming under linear and nonlinear differentiable constraints. A reduced gradient and GRG (generalized reduced gradient) descent methods involving stochastic perturbation are proposed and we give a mathematical result establishing the convergence to a global minimizer. Numerical examples are given in order to show that the method is effective to calculate. Namely, we consider classical tests such as the statistical problem, the octagon problem, the mixture problem and an application to the linear optimal control servomotor problem. (C) 2013 Elsevier Inc. All rights reserved.
This work deals with a Berge equilibrium problem (BEP). Based on the existence results of Berge equilibrium of Nessah et al. (Appl Math Lett 20(8):926-932. 2007), we consider BEP with concave objective functions. The ...
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This work deals with a Berge equilibrium problem (BEP). Based on the existence results of Berge equilibrium of Nessah et al. (Appl Math Lett 20(8):926-932. 2007), we consider BEP with concave objective functions. The existence of Berge equilibrium has been proven. BEP reduces to nonsmooth optimization problem. Then using a regularized function, we reduce a problem of finding Berge equilibrium to a nonconvex global optimization problem with a differentiable objective functions. The later allows to apply optimization methods and algorithms to solve the original problem.
We propose a descent subgradient algorithm for unconstrained nonsmooth nonconvex multiobjective optimization problems. To find a descent direction, we present an iterative process that efficiently approximates the eps...
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We propose a descent subgradient algorithm for unconstrained nonsmooth nonconvex multiobjective optimization problems. To find a descent direction, we present an iterative process that efficiently approximates the epsilon-subdifferential of each objective function. To this end, we develop a new variant of Mifflin's line search in which the subgradients are arbitrary and its finite convergence is proved under a semismooth assumption. To reduce the number of subgradient evaluations, we employ a backtracking line search that identifies the objectives requiring an improvement in the current approximation of the epsilon-subdifferential. Meanwhile, for the remaining objectives, new subgradients are not computed. Unlike bundle-type methods, the proposed approach can handle nonconvexity without the need for algorithmic adjustments. Moreover, the quadratic subproblems have a simple structure, and hence the method is easy to implement. We analyze the global convergence of the proposed method and prove that any accumulation point of the generated sequence satisfies a necessary Pareto optimality condition. Furthermore, our convergence analysis addresses a theoretical challenge in a recently developed subgradient method. Through numerical experiments, we observe the practical capability of the proposed method and evaluate its efficiency when applied to a diverse range of nonsmooth test problems.
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.
In this paper, we present and analyze a finitely convergent disjunctive cutting plane algorithm to obtain an \epsilon -optimal solution or detect the infeasibility of a general nonconvex continuous bilinear program. W...
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In this paper, we present and analyze a finitely convergent disjunctive cutting plane algorithm to obtain an \epsilon -optimal solution or detect the infeasibility of a general nonconvex continuous bilinear program. While the cutting planes are obtained like Saxena, Bonami, and Lee [Math. Prog., the algorithm that guarantees finite convergence is exploring near-optimal extreme point solutions to a current relaxation at each iteration. In this sense, the presented algorithm and its analysis extend the work Owen and Mehrotra [Math. Prog., 89 (2001), pp. 437--448] for solving mixed-integer linear programs to the general bilinear programs.
Hyperspectral image (HSI) and multispectral image (MSI) fusion aims at producing a super-resolution image (SRI). In this paper, we establish a nonconvex optimization model for image fusion problem through low rank ten...
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Hyperspectral image (HSI) and multispectral image (MSI) fusion aims at producing a super-resolution image (SRI). In this paper, we establish a nonconvex optimization model for image fusion problem through low rank tensor triple decomposition. Using the limited memory BFGS (L-BFGS) approach, we develop a first-order optimization algorithm for obtaining the desired super-resolution image (TTDSR). Furthermore, two detailed methods are provided for calculating the gradient of the objective function. With the aid of the KurdykaLojasiewicz property, the iterative sequence is proved to converge to a stationary point. Finally, experimental results on different datasets show the effectiveness of our proposed approach.
The inverse kinematics problem plays a crucial role in robotic manipulator planning, autonomous control, and object grasping. This problem can be solved in simple environments based on existing studies. However, it is...
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The inverse kinematics problem plays a crucial role in robotic manipulator planning, autonomous control, and object grasping. This problem can be solved in simple environments based on existing studies. However, it is still challenging to quickly find a feasible inverse kinematic solution when obstacle avoidance is required. In this paper, we present a nonconvex composite programming method to solve the inverse kinematics problem with overhead obstacle-avoidance requirements. Our method enables efficient obstacle avoidance by directly calculating the minimum distance between the manipulator and the overhead environment. We construct end-effector error functions based on the Product of Exponentials model and explicitly provide their gradient formula. We derive the minimum distance based on the geometry parametric equation and directly utilize it to construct the obstacle avoidance function. We propose an enhanced version of adaptive moment estimation based on shorttime gradient information to improve optimization performance. Finally, we conduct simulations and experiments in overhead line environments. Comparative results with other optimization methods demonstrate that our proposed method achieves a high success rate with a low solution time.
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