Differential stability properties in convex scalar and vector optimization

This paper focuses on formulas for the ε-subdifferential of the optimal value function of scalar and vector convex optimization problems. These formulas can be applied when the set of solutions of the problem is empty. In the scalar case, both unconstrained problems and problems with an inclusion constraint are considered. For the last ones, limiting results are derived, in such a way that no qualification conditions are required. The main mathematical tool is a limiting calculus rule for the ε-subdifferential of the sum of convex and lower semicontinuous functions defined on a (non necessarily reflexive) Banach space. In the vector case, unconstrained problems are studied and exact formulas are derived by linear scalarizations. These results are based on a concept of infimal set, the notion of cone proper set and an ε-subdifferential for convex vector functions due to Taa.

Estimates of objective function minimum for solving linear fractional unconstrained combinatorial optimization problems on arrangements

The paper is devoted to the study of one class of Euclidean combinatorial optimization problems — combinatorial optimization problems on the general set of arrangements with linear fractional objective function and without additional (non-combinatorial) constraints. The paper substantiates the improvement of the polynomial algorithm for solving the specified class of problems. This algorithm foresees solving a finite sequence of linear unconstrained problems of combinatorial optimization on arrangements. The modification of the algorithm is based on the use of estimates of the objective function on the feasible set. This allows to exclude some of the problems from consideration and reduce the number of problems to be solved. The numerical experiments confirm the practical efficiency of the proposed approach.

A New Hybrid Three-Term Conjugate Gradient Algorithm for Large-Scale Unconstrained Problems

Three-term conjugate gradient methods have attracted much attention for large-scale unconstrained problems in recent years, since they have attractive practical factors such as simple computation, low memory requirement, better descent property and strong global convergence property. In this paper, a hybrid three-term conjugate gradient algorithm is proposed and it owns a sufficient descent property, independent of any line search technique. Under some mild conditions, the proposed method is globally convergent for uniformly convex objective functions. Meanwhile, by using the modified secant equation, the proposed method is also global convergence without convexity assumption on the objective function. Numerical results also indicate that the proposed algorithm is more efficient and reliable than the other methods for the testing problems.

An enhanced approach for optimizing mathematical and structural problems by combining PSO, GSA and gradient directions

In this paper, the combination of particle swarm optimization (PSO) and gravitational search algorithm (GSA) is enhanced by the first-order gradient method and a new optimization algorithm is introduced as GPSG. In metaheuristic methods, some search directions are selected at random and the resulting points gradually progress toward the optimal. Since the gradient direction usually has the largest decrease in the desired function, it is added to the GSA and PSO process to allow for faster and more accurate convergence. By integrating the metaheuristic methods with the gradient directions, a powerful method for optimizing functions has been made possible. Numerous examples of unconstrained problems of mathematical functions of CEC2005 and constrained examples of stress and displacement structural design problems have been chosen to demonstrate the reliability and capability of the presented method.

A Proximal Bundle Method with Exact Penalty Technique and Bundle Modification Strategy for Nonconvex Nonsmooth Constrained Optimization

The bundle modification strategy for the convex unconstrained problems was proposed by Alexey et al. [[2007] European Journal of Operation Research, 180(1), 38–47.] whose most interesting feature was the reduction of the calls for the quadratic programming solver. In this paper, we extend the bundle modification strategy to a class of nonconvex nonsmooth constraint problems. Concretely, we adopt the convexification technique to the objective function and constraint function, take the penalty strategy to transfer the modified model into an unconstrained optimization and focus on the unconstrained problem with proximal bundle method and the bundle modification strategies. The global convergence of the corresponding algorithm is proved. The primal numerical results show that the proposed algorithms are promising and effective.

An Improved Grasshopper Optimizer for Global Tasks

The grasshopper optimization algorithm (GOA) is a metaheuristic algorithm that mathematically models and simulates the behavior of the grasshopper swarm. Based on its flexible, adaptive search system, the innovative algorithm has an excellent potential to resolve optimization problems. This paper introduces an enhanced GOA, which overcomes the deficiencies in convergence speed and precision of the initial GOA. The improved algorithm is named MOLGOA, which combines various optimization strategies. Firstly, a probabilistic mutation mechanism is introduced into the basic GOA, which makes full use of the strong searchability of Cauchy mutation and the diversity of genetic mutation. Then, the effective factors of grasshopper swarm are strengthened by an orthogonal learning mechanism to improve the convergence speed of the algorithm. Moreover, the application of probability in this paper greatly balances the advantages of each strategy and improves the comprehensive ability of the original GOA. Note that several representative benchmark functions are used to evaluate and validate the proposed MOLGOA. Experimental results demonstrate the superiority of MOLGOA over other well-known methods both on the unconstrained problems and constrained engineering design problems.