Description:
We present an algorithm to locally minimize nonsmooth, nonconvex functions. In order to find descent directions, the notion of quasisecants, introduced in this paper, is applied. We prove that the algorithm converges to Clarke stationary points. Numerical results are presented demonstrating the applicability of the proposed algorithm to a wide variety of nonsmooth, nonconvex optimization problems. We also compare the proposed algorithm with the bundle method using numerical results.
Description:
Clustering is an important task in data mining. It can be formulated as a global optimization problem which is challenging for existing global optimization techniques even in medium size data sets. Various heuristics were developed to solve the clustering problem. The global k-means and modified global k-means are among most efficient heuristics for solving the minimum sum-of-squares clustering problem. However, these algorithms are not always accurate in finding global or near global solutions to the clustering problem. In this paper, we introduce a new algorithm to improve the accuracy of the modified global k-means algorithm in finding global solutions. We use an auxiliary cluster problem to generate a set of initial points and apply the k-means algorithm starting from these points to find the global solution to the clustering problems. Numerical results on 16 real-world data sets clearly demonstrate the superiority of the proposed algorithm over the global and modified global k-means algorithms in finding global solutions to clustering problems.
Description:
It is well known that apart from the Petersen graph there are no Moore graphs of degree 3. As a cubic graph must have an even number of vertices, there are no graphs of maximum degree 3 and
Description:
In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.
Description:
In this paper, we consider a distributed nonsmooth optimization problem over a computational multi-agent network. We first extend the (centralized) Nesterov’s random gradient-free algorithm and Gaussian smoothing technique to the distributed case. Then, the convergence of the algorithm is proved. Furthermore, an explicit convergence rate is given in terms of the network size and topology. Our proposed method is free of gradient, which may be preferred by practical engineers. Since only the cost function value is required, our method may suffer a factor up to d (the dimension of the agent) in convergence rate over that of the distributed subgradient-based methods in theory. However, our numerical simulations show that for some nonsmooth problems, our method can even achieve better performance than that of subgradient-based methods, which may be caused by the slow convergence in the presence of subgradient.
Description:
In this paper, we consider a distributed nonsmooth optimization problem over a computational multi-agent network. We first extend the (centralized) Nesterov’s random gradient-free algorithm and Gaussian smoothing technique to the distributed case. Then, the convergence of the algorithm is proved. Furthermore, an explicit convergence rate is given in terms of the network size and topology. Our proposed method is free of gradient, which may be preferred by practical engineers. Since only the cost function value is required, our method may suffer a factor up to d (the dimension of the agent) in convergence rate over that of the distributed subgradient-based methods in theory. However, our numerical simulations show that for some nonsmooth problems, our method can even achieve better performance than that of subgradient-based methods, which may be caused by the slow convergence in the presence of subgradient.
Description:
Pumping cost constitutes the main part of the overall operating cost of water distribution systems. There are different optimization formulations of the pumping cost minimization problem including those with application of continuous and integer programming approaches. To date mainly various metaheuristics have been applied to solve this problem. However, the comprehensive comparison of those metaheuristics has not been done. Such a comparison is important to identify strengths and weaknesses of different algorithms which reflects on their performance. In this paper, we present a methodology for comparative analysis of widely used metaheuristics for solving the pumping cost minimization problem. This methodology includes the following comparison criteria: (a) the "optimal solution" obtained; (b) the efficiency; and (c) robustness. Algorithms applied are: particle swarm optimization, artificial bee colony and firefly algorithms. These algorithms were applied to one test problem available in the literature. The results obtained demonstrate that the artificial bee colony is the most robust and the firefly is the most efficient and accurate algorithm for this test problem. Funding :ARC
Description:
This paper presents some applications of the canonical dual theory in optimal control problems. The analytic solutions of several nonlinear and nonconvex problems are investigated by global optimizations. It turns out that the backward differential flow defined by the KKT equation may reach the globally optimal solution. The analytic solution to an optimal control problem is obtained via the expression of the co-state. Some examples are illustrated.
Description:
In this paper, we propose a new algorithm for global minimization of functions represented as a difference of two convex functions. The proposed method is a derivative free method and it is designed by adapting the extended cutting angle method. We present preliminary results of numerical experiments using test problems with difference of convex objective functions and box-constraints. We also compare the proposed algorithm with a classical one that uses prismatical subdivisions.