The ''attack-defense'' model on networks with the initial residuals of the parties

Статья обобщает игру «нападение-оборона», имеющую сетевую структуру, в части учета начальных остатков ресурсов сторон и основана на работе R. Hohzaki, V. Tanaka. В отличие от последней, оборона на каждом из возможных направлений движения между вершинами сети, заданных ориентированными ребрами, может иметь ненулевые начальные остатки ресурсов сторон, что приводит в общем случае к выпуклым минимаксным задачам, которые могут быть решены методом субградиентного спуска. В частности, изучаемая модель обобщает игру «нападение-оборона» с начальными остатками, предложенную В.Ф.Огарышевым, на сетевой случай. The article generalizes the "attack-defense" game with the network structure, in terms of accounting for the initial residuals of the parties' resources and is based on the work by Hohzaki and Tanaka. In contrast to the latter, the defense on each of the possible movement directions between the network’s vertices, given by the oriented edges, can have nonzero initial residuals of the parties' resources, which generally leads to convex minimax problems that can be solved by the subgradient descent method. In particular, the model under study generalizes the "attack-defense" game with initial residuals, proposed by Ogaryshev, to the network case.

A Shuffle-Based Artificial Bee Colony Algorithm for Solving Integer Programming and Minimax Problems

The artificial bee colony (ABC) algorithm is a prominent swarm intelligence technique due to its simple structure and effective performance. However, the ABC algorithm has a slow convergence rate when it is used to solve complex optimization problems since its solution search equation is more of an exploration than exploitation operator. This paper presents an improved ABC algorithm for solving integer programming and minimax problems. The proposed approach employs a modified ABC search operator, which exploits the useful information of the current best solution in the onlooker phase with the intention of improving its exploitation tendency. Furthermore, the shuffle mutation operator is applied to the created solutions in both bee phases to help the search achieve a better balance between the global exploration and local exploitation abilities and to provide a valuable convergence speed. The experimental results, obtained by testing on seven integer programming problems and ten minimax problems, show that the overall performance of the proposed approach is superior to the ABC. Additionally, it obtains competitive results compared with other state-of-the-art algorithms.

A new three-term conjugate gradient method for solving the finite minimax problems

In this paper, we consider the method for solving the finite minimax problems. By using the exponential penalty function to smooth the finite minimax problems, a new three-term nonlinear conjugate gradient method is proposed for solving the finite minimax problems, which generates sufficient descent direction at each iteration. Under standard assumptions, the global convergence of the proposed new three-term nonlinear conjugate gradient method with Armijo-type line search is established. Numerical results are given to illustrate that the proposed method can efficiently solve several kinds of optimization problems, including the finite minimax problem, the finite minimax problem with tensor structure, the constrained optimization problem and the constrained optimization problem with tensor structure.

The ``attack-defense'' game with restrictions on the intake capacity of points

Работа обобщает игру «нападение-оборона» Ю.Б.Гермейера в части учета пропускной способности пунктов и основана на его обобщенном принципе уравнивания, что приводит в случае однородности ресурсов сторон к выпуклым минимаксным задачам, которые могут быть решены методом субградиентного спуска. Классическая модель «нападение-оборона» Ю.Б.Гермейера является модификацией модели О.Гросса. В работе В.Ф. Огарышева исследована игровая модель, обобщающая модели Гросса и Гермейера. В работе Д.А. Молодцова изучалась модель Гросса с непротивоположными интересами сторон, в работах Т.Н.Данильченко, К.К. Масевич и Б.П.Крутова - динамические расширения модели. В военных моделях пункты интерпретируются обычно как направления и характеризуют пространственное распределение ресурсов защиты по ширине. Однако реально имеют место также ограничения по пропускной способности пунктов (направлений). Это приводит в случае однородных ресурсов к минимаксным задачам для определения гарантированного результата (НГР) обороны. Получена точная верхняя оценка для НГР обороны, которая показывает потенциальные возможности обороны с учетом пропускной способности пунктов (направлений). The work generalizes the Germeier’s "attack-defense" game in terms of accounting for the intake capacity of points and is based on his generalized equalization principle, which leads to convex minimax problems that can be solved by subgradient descent in the case of homogeneity of the parties' resources. The classical Germeier’s "attack-defense" model is a modification of the Gross’ model. The game model that generalizes Gross’ model and Germeier’s model was studied by Ogaryshev. Molodtsov studied the Gross’s model with nonantagonistic interests of the parties; Danilchenko, Masevich and Krutova studied the dynamic extensions of the model. In the military models the points are usually interpreted as directions and characterize the spatial distribution of defense resources by width. However, there are also actual restrictions on the intake capacity of points. This leads, in the case of homogeneous resources, to minimax problems for determining the best guaranteed defense result (BGDR). An accurate upper estimate for the best guaranteed defense result was obtained, which shows the potential defense capabilities taking into account the intake capacity of points.

Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results.