A new modified three-term conjugate gradient (CG) method is shown for solving the large scale optimization problems. The idea relates to the famous Polak-Ribiere-Polyak (PRP) formula. As the numerator of PRP plays a v...
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A new modified three-term conjugate gradient (CG) method is shown for solving the large scale optimization problems. The idea relates to the famous Polak-Ribiere-Polyak (PRP) formula. As the numerator of PRP plays a vital role in numerical result and not having the jamming issue, PRP method is not globally convergent. So, for the new three-term CG method, the idea is to use the PRP numerator and combine it with any good CG formula's denominator that performs well. The new modification of three-term CG method possesses the sufficient descent condition independent of any line search. The novelty is that by using the Wolfe Powell line search the new modification possesses global convergence properties with convex and nonconvex functions. Numerical computation with the Wolfe Powell line search by using the standard test function of optimization shows the efficiency and robustness of the new modification.
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.
In this paper we consider the problem of clustering m objects into c clusters. The objects are represented by points in an n-dimensional Euclidean space, and the objective is to classify these m points into c clusters...
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In this paper we consider the problem of clustering m objects into c clusters. The objects are represented by points in an n-dimensional Euclidean space, and the objective is to classify these m points into c clusters such that the distance between points within a cluster and its center (which is to be found) is minimized. The problem is a nonconvex program that has many local minima. It has been studied by many researchers and the most well-known algorithm for solving it is the k-means algorithm. In this paper, we develop a new algorithm for solving this problem based on a tabu search technique. Preliminary computational experience on the developed algorithm are encouraging and compare favorably with both the k-means and the simulated annealing algorithms.
We study the geometric evolution of a nonconvex stone by the wearing process via the partial differential equation methods. We use the so-called level set approach to this geometric evolution of a set. We establish a ...
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We study the geometric evolution of a nonconvex stone by the wearing process via the partial differential equation methods. We use the so-called level set approach to this geometric evolution of a set. We establish a comparison theorem, an existence theorem, and some stability properties of solutions of the partial differential equation arising in this level set approach, and define the flow of a set by the wearing process via the level set approach.
We consider two different variational models of transport networks: the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford-Shah image inpainting and techniques de...
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We consider two different variational models of transport networks: the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford-Shah image inpainting and techniques developed in that field, we show for a two-dimensional situation that both highly non-convex network optimization tasks can be transformed into a convex variational problem, which may be very useful from analytical and numerical perspectives. As applications of the convex formulation, we use it to perform numerical simulations (to our knowledge this is the first numerical treatment of urban planning), and we prove a lower bound for the network cost that matches a known upper bound (in terms of how the cost scales in the model parameters) which helps better understand optimal networks and their minimal costs.
In this paper, we propose a feasible-direction method for large-scale nonconvex programs, where the gradient projection on a linear subspace defined by the active constraints of the original problem is determined by d...
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In this paper, we propose a feasible-direction method for large-scale nonconvex programs, where the gradient projection on a linear subspace defined by the active constraints of the original problem is determined by dual decomposition. Results are extended for dynamical problems which include distributed delays and constraints both in state and control variables. The approach is compared with other feasible-direction approaches, and the method is applied to a power generation problem. Some computational results are included.
This paper considers the problem of choosingn numbers in the unit interval in such a way that their products in pairs are distributed as evenly as possible. Specifically, it is desired (i) to maximize the minimum diff...
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This paper considers the problem of choosingn numbers in the unit interval in such a way that their products in pairs are distributed as evenly as possible. Specifically, it is desired (i) to maximize the minimum difference between successive products or (ii) to minimize the maximum difference between successive products. These problems are solved for the casesn=1, 2, 3. For generaln, the problems are solved under certain additional restrictions, and the limiting behavior for largen is determined. The situation for generaln is investigated further by posing and solving a continuous analogue of the discrete problem. This leads to a heuristic method of determining the optimum order of products in the general case.
In order to restore the high quality image, we propose a compound regularization method which combines a new higher-order extension of total variation (TV+TV2) and a nonconvex sparseness-inducing penalty. Considering ...
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In order to restore the high quality image, we propose a compound regularization method which combines a new higher-order extension of total variation (TV+TV2) and a nonconvex sparseness-inducing penalty. Considering the presence of varying directional features in images, we employ the shearlet transform to preserve the abundant geometrical information of the image. The nonconvex sparseness-inducing penalty approach increases robustness to noise and image nonsparsity. In what follows, we present the numerical solution of the proposed model by employing the split Bregman iteration and a novel p-shrinkage operator. And finally, we perform numerical experiments for image denoising, image deblurring, and image reconstructing from incomplete spectral samples. The experimental results demonstrate the efficiency of the proposed restoration method for preserving the structure details and the sharp edges of image.
During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exh...
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During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter-examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity.
A deterministic algorithm for solving nonconvex NLPs globally using a reduced-space approach is presented. These problems are encountered when real-world models are involved as nonlinear equality constraints and the d...
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A deterministic algorithm for solving nonconvex NLPs globally using a reduced-space approach is presented. These problems are encountered when real-world models are involved as nonlinear equality constraints and the decision variables include the state variables of the system. By solving the model equations for the dependent (state) variables as implicit functions of the independent (decision) variables, a significant reduction in dimensionality can be obtained. As a result, the inequality constraints and objective function are implicit functions of the independent variables, which can be estimated via a fixed-point iteration. Relying on the recently developed ideas of generalized McCormick relaxations and McCormick-based relaxations of algorithms and subgradient propagation, the development of McCormick relaxations of implicit functions is presented. Using these ideas, the reduced space, implicit optimization formulation can be relaxed. When applied within a branch-and-bound framework, finite convergence to epsilon-optimal global solutions is guaranteed.
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