Optimization Letters最新文献

In inverse optimization problems, we are given a feasible solution to an underlying optimization problem, and the goal is to modify the problem parameters so that the given input solution becomes optimal. In the minimum-cost setting, the underlying optimization problem is endowed with a linear cost function, and the goal is to modify the costs by a small deviation vector so that the input solution becomes optimal. The difference between the new and the original cost functions can be measured in several ways. In this paper, we focus on two objectives: the weighted bottleneck Hamming distance and the weighted ${\ell }_{\infty }$ -norm. We consider a general model in which the coordinates of the deviation vector are required to fall within given lower and upper bounds. For the weighted bottleneck Hamming distance objective, we present a simple, purely combinatorial algorithm that determines an optimal deviation vector in strongly polynomial time. For the weighted ${\ell }_{\infty }$ -norm objective, we give a min-max characterization for the optimal solution, and provide a pseudo-polynomial algorithm for finding an optimal deviation vector that runs in strongly polynomial time in the case of unit weights. For both objectives, we assume that an algorithm with the same time complexity for solving the underlying combinatorial optimization problem is available.

We study a method that involves principally convex feasibility-seeking and makes secondary efforts of objective function value reduction. This is the well-known superiorization method (SM), where the iterates of an asymptotically convergent iterative feasibility-seeking algorithm are perturbed by objective function nonascent steps. We investigate the question under what conditions a sequence generated by an SM algorithm asymptotically converges to a feasible point whose objective function value is superior (meaning smaller or equal) to that of a feasible point reached by the corresponding unperturbed one (i.e., the exactly same feasibility-seeking algorithm that the SM algorithm employs.) This question is yet only partially answered in the literature. We present a condition under which an SM algorithm that uses negative gradient descent steps in its perturbations fails to yield such a superior outcome. The significance of the discovery of this "negative condition" is that it necessitates that the inverse of this condition will have to be assumed to hold in any future guarantee result for the SM. The condition is important for practitioners who use the SM because it is avoidable in experimental work with the SM, thus increasing the success rate of the method in real-world applications.

We study a single machine scheduling problem with generalized due-dates and general position-dependent job processing times. The objective function is minimum number of tardy jobs. The problem is proved to be NP-hard in the strong sense. We introduce an efficient algorithm that solves medium size problems in reasonable running time. A simple and efficient heuristic is also introduced, which obtained the optimal solution in the vast majority of our tests.

In this paper, we consider the numerical solution of a class of vertical tensor complementarity problems. By reformulating the involved vertical tensor complementarity problem (VTCP) as an equivalent projected fixed point equation, together with the relevant properties of the power Lipschitz tensor, we propose a projected fixed point method for the involved VTCP, and discuss its convergence properties. Numerical experiments are given to illustrate the effectiveness of the proposed method.

We are given a set of indivisible goods and a set of m agents where each good has a size and each agent has an additive valuation function and a budget. The budgeted maximin share allocation problem is to find a feasible allocation such that the size of the bundle allocated to each agent does not exceed its budget, and the minimum ratio of the valuation and the maximin share (MMS) value of any agent is as large as possible, where the MMS value of each agent is that he can achieve by dividing the goods into n bundles, and receiving his least desirable bundle. In this paper, we prove the existence of (frac{n}{3n-2})-approximate MMS allocation and give an instance which does not have a ((frac{3}{4}+epsilon ))-approximate MMS allocation, for any (epsilon in (0,1)). Moreover, we provide a polynomial time algorithm to find an (frac{1}{3})-MMS allocation, and prove that there is no ((frac{2}{3} + epsilon ))-approximate algorithm in polynomial time unless (mathcal{P}=mathcal{N}mathcal{P}).

We introduce and study some explicit iterative algorithms for solving the system of split equality problems in Hilbert spaces. The strong convergence of the proposed algorithms is proved by using some milder conditions put on control parameters than the one used in Tuyen (Bull Malays Math Sci Soc 46:44, 2023).

We establish the optimal ergodic convergence rate for the classical projected subgradient method with time-varying step-sizes. This convergence rate remains the same even if we slightly increase the weight of the most recent points, thereby relaxing the ergodic sense.

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.

The weighted linear complementarity problem (wLCP) can be used for modelling a large class of problems from science and economics. In this paper, we introduce a new class of weighted complementarity functions and show that it is continuously differentiable everywhere. By using this function, we propose a two steps Levenberg–Marquardt-type method to solve the wLCP. Under suitable conditions, we prove that the proposed method is globally convergent and the generated iteration sequence is bounded. Moreover, we show that the proposed method has cubic convergence rate under the local error bound condition. Some numerical results are reported.

In this paper, we present a Levenberg–Marquardt algorithm for nonlinear equations, where the exact Jacobians are unavailable, but their model approximations can be built in some random fashion. We study the complexity of the algorithm and show that the upper bound of the iteration numbers in expectation to obtain a first order stationary point is (O(epsilon ^{-3})).