Coevolutionary Framework for Generalized Multimodal Multi-Objective Optimization

Most multimodal multi-objective evolutionary algorithms(MMEAs)aim to find all global Pareto optimal sets(PSs)for a multimodal multi-objective optimization problem(MMOP).However,in real-world problems,decision makers(DMs)may be also interested in local PSs.Also,searching for both global and local PSs is more general in view of dealing with MMOPs,which can be seen as generalized MMOPs.Moreover,most state-of-theart MMEAs exhibit poor convergence on high-dimension MMOPs and are unable to deal with constrained MMOPs.To address the above issues,we present a novel multimodal multiobjective coevolutionary algorithm(Co MMEA)to better produce both global and local PSs,and simultaneously,to improve the convergence performance in dealing with high-dimension MMOPs.Specifically,the Co MMEA introduces two archives to the search process,and coevolves them simultaneously through effective knowledge transfer.The convergence archive assists the Co MMEA to quickly approach the Pareto optimal front.The knowledge of the converged solutions is then transferred to the diversity archive which utilizes the local convergence indicator and the-dominance-based method to obtain global and local PSs effectively.Experimental results show that Co MMEA is competitive compared to seven state-of-the-art MMEAs on fifty-four complex MMOPs.

Distributed Optimal Variational GNE Seeking in Merely Monotone Games

In this paper, the optimal variational generalized Nash equilibrium(v-GNE) seeking problem in merely monotone games with linearly coupled cost functions is investigated, in which the feasible strategy domain of each agent is coupled through an affine constraint. A distributed algorithm based on the hybrid steepest descent method is first proposed to seek the optimal v-GNE. Then, an accelerated algorithm with relaxation is proposed and analyzed, which has the potential to further improve the convergence speed to the optimal v-GNE. Some sufficient conditions in both algorithms are obtained to ensure the global convergence towards the optimal v-GNE. To illustrate the performance of the algorithms, numerical simulation is conducted based on a networked Nash-Cournot game with bounded market capacities.

QUASI-EQUILIBRIUM PROBLEMS AND CONSTRAINED MULTIOBJECTIVE GAMES IN GENERALIZED CONVEX SPACE

A class of quasi-equilibrium problems and a class of constrained multiobjective games were introduced and studied in generalized convex spaces without linear structure. First, two existence theorems of solutions for quasi-equilibrium problems are proved in noncompact generalized convex spaces. Then, ar applications of the quasi-equilibrium existence theorem, several existence theorems of weighted Nash-equilibria and Pareto equilibria for the constrained multiobjective games are established in noncompact generalized convex spaces. These theorems improve, unify, and generalize the corresponding results of the multiobjective games in recent literatures.

Aircraft Landing Gear Control with Multi-Objective Optimization Using Generalized Cell Mapping

This paper presents a numerical algorithm tuning aircraft landing gear control system with three objectives,including reducing relative vibration, reducing hydraulic strut force and controlling energy consumption. Sliding mode control is applied to the vibration control of a simplified landing gear model with uncertainty. A two-stage generalized cell mapping algorithm is applied to search the Pareto set with gradient-free scheme. Drop test simulations over uneven runway show that the vibration and force interaction can be considerably reduced, and the Pareto optimum form a tight range in time domain.

NONCOMPACT INFINITE OPTIMIZATION AND EQUILIBRIA OF CONSTRAINED GAMES IN GENERALIZED CONVEX SPACES

By applying a new existence theorem of quasi-equilibrium problems due to the author, some existence theorems of solutions for noncompact infinite optimization problems and noncompact constrained game problems are proved in generalized convex spaces without linear structure. These theorems improve and generalize a number of important results in recent literature.

Generalized constrained multiobjective games in locally FC-uniform spaces

A new class of generalized constrained multiobjective games is introduced and studied in locally FC-uniform spaces without convexity structure where the number of players may be finite or infinite and all payoff functions get their values in an infinite-dimensional space. By using a Himmelberg type fixed point theorem in locally FC-uniform spaces due to author, some existence theorems of weak Paxeto equilibria for the generalized constrained multiobjective games are established in locally FC-uniform spaces. These theorems improve, unify and generalize the corresponding results in recent literatures.

EQUILIBIA OF NONCOMPACT GENERALIZED GAMES WITH CORRESPONDENCES WITHOUT OPEN LOWER SECTIONS

The new classes of Fc-correspondences and Fc -majorised correspondenceswithout open lower sections is introduced. Some existence theorems of madximal elements ofthe Fc -correspondences and the Fc-majorized correspondences defined on noncompactset in topological vector spaces are obtained. As applications, some existence theoremsof equilibrium points for one-person games, qualitative games and generalized games withthe Fc -majorized correspondences devined on noncompact strategy sets in topologicalvector spaces are also given. These theorems improve and generalize several known resultsin recent literature.

EQUILIBRIUM EXISTENCE FOR MULTI-LEADER-FOLLOWER GENERALIZED CONSTRAINED MULTIOBJECTIVE GAMES IN LOCALLY FC-UNIFORM SPACES

In this article, we introduce and study some new classes of multi-leader-follower generalized constrained multiobjective games in locally FC-uniform spaces where the number of leaders and followers may be finite or infinite and the objective functions of the followers obtain their values in infinite-dimensional spaces. Each leader has a constrained correspondence. By using a collective fixed point theorem in locally FC-uniform spaces due to author, some existence theorems of equilibrium points for the multi-leader-follower generalized constrained multiobjective games are established under nonconvex settings. These results generalize some corresponding results in recent literature.

Maximal elements and generalized games involving condensing mappings in locally FC-uniform spaces and applications(Ⅰ)

First, the notions of the measure of noncompactness and condensing setvalued mappings are introduced in locally FC-uniform spaces without convexity structure. A new existence theorem of maximal elements of a family of set-valued mappings involving condensing mappings is proved in locally FC-uniform spaces. As applications, some new equilibrium existence theorems of generalized game involving condensing mappings are established in locally FC-uniform spaces. These results improve and generalize some known results in literature to locally FC-uniform spaces. Some further applications of our results to the systems of generalized vector quasi-equilibrium problems will be given in a follow-up paper.

MULTI-OBJECTIVE SHAPE DESIGN IN AERODYNAMICS USING GAME STRATEGY

Multi-objective optimization for the optimum shape design is introduced in aerodynamics using the Game theory. Based on the control theory, the employed optimizer and the negative feedback are used to implement the constraints. All the constraints are satisfied implicitly and automatically in the design. Furthermore,the above methodology is combined with a formulation derived from the Game theory to treat multi-point airfoil optimization. Airfoil shapes are optimized according to various aerodynamics criteria. In the symmetric Nash game, each “player” is responsible for one criterion, and the Nash equilibrium provides a solution to the multipoint optimization. Design results confirm the efficiency of the method.

Multi-objective optimization strategies using adjoint method and game theory in aerodynamics

There are currently three different game strategies originated in economics： （1） Cooperative games （Pareto front）, （2） Competitive games （Nash game） and （3） Hierarchical games （Stackelberg game）. Each game achieves different equilibria with different performance, and their players play different roles in the games. Here, we introduced game concept into aerodynamic design, and combined it with adjoint method to solve multi- criteria aerodynamic optimization problems. The performance distinction of the equilibria of these three game strategies was investigated by numerical experiments. We computed Pareto front, Nash and Stackelberg equilibria of the same optimization problem with two conflicting and hierarchical targets under different parameterizations by using the deterministic optimization method. The numerical results show clearly that all the equilibria solutions are inferior to the Pareto front. Non-dominated Pareto front solutions are obtained, however the CPU cost to capture a set of solutions makes the Pareto front an expensive tool to the designer.

The Coalition Cooperative Game Method and Its Application in Multi-objective Optimization Design

This paper proposes a multi-objective optimization design method based on the coalition cooperative game theory where the three design goals have been seen as three game players. By calculating the affecting factors and fuzzy clustering, the design variables are divided into different strategic spaces which belong to each player, then it constructs a payoff function based on the coalition mechanism. Each game player takes its own revenue function as a target and obtains the best strategy versus other players. The best strategies of all players consist of the strategy permutation of a round game and it obtains the final game solutions through multi-round games according to the convergence criterion. A multi-objective optimization example of the luff mechanism of compensative sheave block shows the effectiveness of the coalition cooperative game method.

The Multi-Objective Optimization of AFPM Generators with Double-Sided Internal Stator Structures for Vertical Axis Wind Turbines

The axial flux permanent magnet(AFPM)generator with double-sided internal stator structure is highly suitable for vertical axis wind turbines due to its high power density.The performance of the AFPM generator with double-sided internal stator structure can be improved by the reasonable design of electromagnetic parameters.To further improve the overall performance of the AFPM generator with double-sided internal stator structure,multivariable(coil widthω_(c),permanent magnet thickness h,pole arc coefficient α_(p) and working air gap l_(g))and multi-objective(generator efficiencyη,total harmonic distortion of the voltage THD and induced electromotive force amplitude EMF)functional relationships are innovatively established.Orthogonal analysis,mean analysis and variance analysis are performed on the influence parameters by combining the Taguchi method and response surface methodology to study the influence degrees of each influence parameter on the optimization objectives to determine the most appropriate electromagnetic parameters.The optimization results are verified by 3D finite element analysis.The optimized APFM generator with double-sided internal stator structure exhibits superior economy,stronger magnetic density,higher efficiency and improved power quality.

The Research of Examination Paper Generation Based on Index System Metrics and Multi-Objective Strategy

Since the examination paper generated with computer by the algorithms of random and backtracking takes on inferior quality and inefficient, and the question of generating examination paper with computer has the character of multi-ob-jective because of the index system metrics, the genetic algorithm with multi-objective strategy optimization is proposed to solve this problem. Mapping the index system to multi-objective functions and optimizing the computing with multi-objective strategy are employed in the algorithm. The genetic algorithm experiment based on the multi-objective strategy optimization shows that the result has the advantages getting tradeoff between performance and quality, and having the ability to tune the performance and quality to meet the user’s requirements.

SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS ON PRODUCT FC-SPACES

By applying a maximal element theorem on product FC-space due to author, some new equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in FC-spaces. By using these equilibrium existence theorems, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are established in noncompact product FC-spaces. These results improve and generalize some recent results in literature to product FC-spaces without any convexity structure.

Multi-objective optimization for voltage and frequency control of smart grids based on controllable loads

The output uncertainty of high-proportion distributed power generation severely affects the system voltage and frequency.Simultaneously,controllable loads have also annually increased,which markedly improve the capability for nodal-power control.To maintain the system frequency and voltage magnitude around rated values,a new multi-objective optimization model for both voltage and frequency control is proposed.Moreover,a great similarity between the multiobjective optimization and game problems appears.To reduce the strong subjectivity of the traditional methods,the idea and method of the game theory are introduced into the solution.According to the present situational data and analysis of the voltage and frequency sensitivities to nodal-power variations,the design variables involved in the voltage and frequency control are classified into two strategy spaces for players using hierarchical clustering.Finally,the effectiveness and rationality of the proposed control are verified in MATLAB.

Multi-objective reentry trajectory optimization method via GVD for hypersonic vehicles

In the constrained reentry trajectory design of hypersonic vehicles, multiple objectives with priorities bring about more difficulties to find the optimal solution. Therefore, a multi-objective reentry trajectory optimization (MORTO) approach via generalized varying domain (GVD) is proposed. Using the direct collocation approach, the trajectory optimization problem involving multiple objectives is discretized into a nonlinear multi-objective programming with priorities. In terms of fuzzy sets, the objectives are fuzzified into three types of fuzzy goals, and their constant tolerances are substituted by the varying domains. According to the principle that the objective with higher priority has higher satisfactory degree, the priority requirement is modeled as the order constraints of the varying domains. The corresponding two-side, single-side, and hybrid-side varying domain models are formulated for three fuzzy relations respectively. By regulating the parameter, the optimal reentry trajectory satisfying priorities can be achieved. Moreover, the performance about the parameter is analyzed, and the algorithm to find its specific value for maximum priority difference is proposed. The simulations demonstrate the effectiveness of the proposed method for hypersonic vehicles, and the comparisons with the traditional methods and sensitivity analysis are presented.

GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMSIN LOCALLY G-CONVEX SPACES

Some classes of generalized vector quasi-equilibrium problems ( in short, GVQEP) are introduced and studied in locally G-convex spaces which includes most of generalized vector equilibrium problems; generalized vector variational inequality problems, quasi-equilibrium problems and quasi-variational inequality problems as special cases. First, an equilibrium existence theorem for one person games is proved in locally G-convex spaces.. As applications, some new existence theorems of solutions for the GVQEP are established in noncompact locally G-convex spaces. These results and argument methods are new and completely different from that in recent literature.

On a Generic Security Game Model

To protect the systems exposed to the Internet against attacks, a security system with the capability to engage with the attacker is needed. There have been attempts to model the engagement/interactions between users, both benign and malicious, and network administrators as games. Building on such works, we present a game model which is generic enough to capture various modes of such interactions. The model facilitates stochastic games with imperfect information. The information is imperfect due to erroneous sensors leading to incorrect perception of the current state by the players. To model this error in perception distributed over other multiple states, we use Euclidean distances between the outputs of the sensors. We build a 5-state game to represent the interaction of the administrator with the user. The states correspond to 1) the user being out of the system in the Internet, and after logging in to the system;2) having low privileges;3) having high privileges;4) when he successfully attacks and 5) gets trapped in a honeypot by the administrator. Each state has its own action set. We present the game with a distinct perceived action set corresponding to each distinct information set of these states. The model facilitates stochastic games with imperfect information. The imperfect information is due to erroneous sensors leading to incorrect perception of the current state by the players. To model this error in perception distributed over the states, we use Euclidean distances between outputs of the sensors. A numerical simulation of an example game is presented to show the evaluation of rewards to the players and the preferred strategies. We also present the conditions for formulating the strategies when dealing with more than one attacker and making collaborations.