Vector Minimax Research Articles

Optimization of spatial arrangement of magnetic field sensors of closed loop system of overhead power lines magnetic field active silencing

Aim. Development of a method for optimization of spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active silencing canceling of the magnetic field generated by overhead power lines. Methodology. Spatial arrangement and angular position of magnetic field sensors of closed loop system of overhead power lines magnetic field active silencing determined based on binary preference relations of local objective for multi-objective minimax optimization problem, in whith the vector objective function calculated based on Biot–Savart law. The solution of this vector minimax optimization problem calculated based on nonlinear Archimedes algorithm of multi-swarm multi-agent optimization. Results. Results of simulation and experimental research of optimal spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active silencing of the magnetic field generated by overhead power lines with a barrel-type arrangement of wires. Originality. The method for optimization of spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active shielding of the magnetic field generated by overhead power lines is developed. Practical value. An important practical problem optimization of spatial arrangement and angular position of magnetic field sensors of a closed system to ensure maximum efficiency of active silencing of the magnetic field generated by overhead power lines has been solved.

The method of multi-objective parametric design of magnetic field active canceling robust system for residential multy-story buildings closed to double-circuit overhead power lines

Aim. Development the method of multi-objective parametric design for robust system of active canceling of magnetic field based on binary preference relations of local objective for multi-objective minimax optimization problem. Methodology. Spatial location coordinates of the compensating winding and the current in the shielding winding were determined during the preference-based multi-objective parametric design of systems of active canceling based on solution of the vector minimax optimization, in whith the vector objective function calculated based on Biot-Savart's law. The solution of this vector minimax optimization problem calculated based on nonlinear Archimedes algorithm. Components of Jacobi matrix and Hesse matrix calculated based on multi-swarm multi-agent optimization. Results. Theoretically and experimentally confirmed the effectiveness of reducing the level of the magnetic field in residential multy-storey old building of a double-circuit overhead power transmission lines with a barrel-type arrangement of wires by means of active shielding with two compensation winding. Originality. The method of multi-objective parametric design for robust system of active canceling of magnetic field based on binary preference relations of local objective for multi-objective minimax optimization problem is developed. Practical value. It is shown the possibility to reduce the level of magnetic field in residential multy-storey old building closed to double-circuit overhead power transmission lines with a barrel-type arrangement of wires by means of system of active canceling with two canceling winding to a level safe for the population with an induction of 0.5 μT.

Hierarchical Population Game Models of Coevolution in Multi-Criteria Optimization Problems under Uncertainty

The article develops hierarchical population game models of co-evolutionary algorithms for solving the problem of multi-criteria optimization under uncertainty. The principles of vector minimax and vector minimax risk are used as the basic principles of optimality for the problem of multi-criteria optimization under uncertainty. The concept of equilibrium of a hierarchical population game with the right of the first move is defined. The necessary conditions are formulated under which the equilibrium solution of a hierarchical population game is a discrete approximation of the set of optimal solutions to the multi-criteria optimization problem under uncertainty.

Vector minimax concave penalty for sparse representation

This paper proposes vector minimax concave (VMC) penalty for sparse representation using tools of Moreau envelope. The VMC penalty is a weighted MC function; by fine tuning the weight of the VMC penalty with given strategy, the VMC regularized least squares problem shares the same global minimizers with the L0 regularization problem but has fewer local minima. Facilitated by the alternating direction method of multipliers (ADMM), the VMC regularization problem can be tackled as a sequence of convex sub-problems, each of which can be solved fast. Theoretical analysis of ADMM shows that the convergence of solving the VMC regularization problem is guaranteed. We present a series of numerical experiments demonstrating the superior performance of the VMC penalty and the ADMM algorithm in broad applications for sparse representation, including sparse denoising, sparse deconvolution, and missing data estimation.

A set optimization approach to zero-sum matrix games with multi-dimensional payoffs

A new solution concept for two-player zero-sum matrix games with multi-dimensional payoffs is introduced. It is based on extensions of the vector order in $$\mathbb {R}^d$$ to order relations in the power set of $$\mathbb {R}^d$$ , so-called set relations, and strictly motivated by the interpretation of the payoff as multi-dimensional loss for one and gain for the other player. The new concept provides coherent worst case estimates for games with multi-dimensional payoffs. It is shown that–in contrast to games with one-dimensional payoffs–the corresponding strategies are different from equilibrium strategies for games with multi-dimensional payoffs. The two concepts are combined into new equilibrium notions for which existence theorems are given. Relationships of the new concepts to existing ones such as Shapley and vector equilibria, vector minimax and maximin solutions as well as Pareto optimal security strategies are clarified.

Postoptimal analysis of a vector minimax combinatorial problem

This paper considers a vector combinatorial problem with minimax criteria that provide the greatest uniformity of the parameters of efficient solutions. The necessary and sufficient conditions are obtained for five well-known types of stability of the problem against perturbations of parameters of the vector objective function.

Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces

In the paper, the notions of generalized loose (weak) saddle points for set-valued mappings are introduced. By applying the fixed point theorem and the scalarization method, some new existence theorems for generalized loose (weak) saddle points are established in locally G -convex space. As applications, some vector minimax theorems are given also.

Vector Minimax from the Viewpoints of the Opponents

The article examines the relationships between multi-criterion minimax values from the viewpoints of a maximizing and a minimizing decision maker. Equality of these values is possible only if the solution of the vector minimax problem is equivalent to the solution of a set of scalar minimax problems.

Recession maps and applications

This paper is devoted to the study of recession maps of set-valued maps in infinite dimensional spaces. Properties and calculus rules of recession maps are provided. As an application, a general closed image theorem is established in a simple way. Some aspects of vector optimization such as the domination property, stability are considered. Vector minimax problems and saddlepoint results are obtained without compactness assumptions