The Uniqueness of Major Efficient Solution for Multiobjective Programming Problem

1　 IntroductionWe consider the multiobjective programming problem:V-minx∈ Xf(x)… (VMP) whererestricted set X Rn,vector objective function f∶X→Rm(m 2 ) .The concept of major efficient solution andα-major efficient solution were firstintroduced by HU Yu-da[1 ] .Moreover,HU Yu-da proved the results thatmajor efficiencysolution setto(VMP) must be a single point set if the vector object function is a strictlyconvex vector function and there exists a major efficient solution in Refs.[1…     

Addressing equity issue in multi-actor policymaking via a system-of-systems approach: Aviation emissions reduction case study

Finding equitable policy solutions is critical for developing sustainable energy use.This paper presents a system-of-systems(SoS) formalism for addressing the equity issue in multi-actor policymaking.In a SoS,the control of the overall system performance is shared among a network of actors.In contrast to a single optimal solution that aggregates objectives of actors,the solution concept of iso-performance is formulated and employed to illuminate multiple solutions and hence the ’space’ for actors to compromise.By specifically accounting for the equity issue,the level of sacrifice each actor makes for each iso-performance solution is computed.To demonstrate the approach,a case study is presented about policymaking to reduce fuel life cycle aviation emissions in the United States based on the year 2020 reduction target,involving government,airlines,jet fuel refinery companies,and aircraft and engine manufacturers.A resource allocation mixed integer programming model is employed to calculate carbon emissions resulting from airlines’ deployment of aircraft fleet to meet changing air transport demand.The paper discusses three iso-performance solutions;each of them requires a different level of sacrifice from each actor.Such an insight can inform policymaking in determining the magnitude of compensation required when a particular solution is pursued.     

Approximation methods of mixed l_1/H_2 optimization problems for MIMO discrete-time systems

The mixed l1/H2 optimization problem for MIMO (multiple input-multiple output) discrete-time systems is considered. This problem is formulated as minimizing the l1-norm of a closed-loop transfer matrix while maintaining the H2-norm of another closed-loop transfer matrix at prescribed level. The continuity property of the optimal value in respect to changes in the H2-norm constraint is studied. The existence of the optimal solutions of mixed l1/H2 problem is proved. Because the solution of the mixed l1/H2 problem is based on the scaled-Q method, it avoids the zero interpolation difficulties. The convergent upper and lower bounds can be obtained by solving a sequence of finite dimensional nonlinear programming for which many efficient numerical optimization algorithms exist.     

THE COMPARISON-NUMBER METHOD FOR FINDING MAJOR OPTIMAL SOLUTIONS AND MAJOR EFFICIENT SOLUTIONS OF THE DISCRETE MULTIOBJECTIVE PROGRAMMING

In this paper, we introduce the comparison number for one feasible solutioncompared with another. With the help of it the comparison-number method for find-ing the major optimal solutions aud major efficient solutions to discrete multiobjectiveprogramming is given.     

CHARACTERIZATIONS OF PROPER EFFICIENT SOLUTIONS IN TERMS OF SADDLE POINTS AND DUALITY

In this paper we characterize the proper efficient solution of a mul-tiobjective programming problem(P)in terms of the saddle point criterion of anew problem(S)and establish a pair of dual problems(S_α)and(D_α)in order todecide the proper efficient solution by the strong duality.We also give methodof characterization in terms of the saddle point criterion of(P).Finally,we provethat every efficient solution of the matrix linear programming problem is a properefficient solution.     

Multiple optimal solutions to a sort of nonlinear optimization problem

The optimization problem is considered in which the objective function is pseudolinear（both pseudoconvex and pseudoconcave） and the constraints are linear. The general expression for the optimal solutions to the problem is derived with the representation theorem of polyhedral sets, and the uniqueness condition of the optimal solution and the computational procedures to determine all optimal solutions （if the uniqueness condition is not satisfied ） are provided. Finally, an illustrative example is also given.     

NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

In this paper, the authors propose Neumann series neural operator(NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions.Helmholtz equation is a crucial partial differential equation(PDE) with applications in various scientific and engineering fields. However, efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber. Recently, deep learning has shown great potential in solving PDEs ...     

VerifyRealRoots:A Matlab Package for Computing Verified Real Solutions of Polynomials Systems of Equations and Inequalities

VerifyRealRoots is a Matlab package for computing and verifying real solutions of polynomial systems of equations and inequalities. It calls Bertini or MMCRSolver for finding approximate real solutions and then applies AINLSS to verify the existence of a regular solution of a polynomial system or applies AINLSS2(AIVISS) to verify the existence of a double solution(a singular solution of an arbitrary multiplicity) of a slightly perturbed polynomial system.     

Novel high-performance element in the electromagnetic finite-element method--node-edge element

It is known in the computational electromagnetics （CEM） that the node element has a relative wellconditioned matrix, but suffers from the spurious solution problem; whereas the edge element has no spurious solutions, but usually produces an ill-conditioned matrix. Particularly, when the mesh is over dense, the iterative solution of the matrix equation from edge element converges very slowly. Based on the node element and edge element, a node-edge element is presented, which has no spurious solutions and better-conditioned matrix. Numerical experiments demonstrate that the proposed node-edge element is more efficient than now-widely used edge element.     

Efficient multiuser detector based on box-constrained deregularization and its FPGA design

Multiuser detection can be described as a quadratic optimization problem with binary constraint.Many techniques are available to find approximate solution to this problem.These techniques can be characterized in terms of complexity and detection performance.The "efficient frontier" of known techniques include the decision-feedback,branch-and-bound and probabilistic data association detectors.The presented iterative multiuser detection technique is based on joint deregularized and box-constrained solution to quadratic optimization with iterations similar to that used in the nonstationary Tikhonov iterated algorithm.The deregularization maximizes the energy of the solution,this is opposite to the Tikhonov regularization where the energy is minimized.However,combined with box-constraints,the deregularization forces the solution to be close to the binary set.We further exploit the boxconstrained dichotomous coordinate descent (DCD) algorithm and adapt it to the nonstationary iterative Tikhonov regularization to present an efficient detector.As a result,the worst-case and average complexity are reduced down to 2.8 and 2.5 floating point operation per second,respectively.The development improves the "efficient frontier" in multiuser detection,which is illustrated by simulation results.Finally,a field programmable gate array (FPGA) design of the detector is presented.The detection performance obtained from the fixed-point FPGA implementation shows a good match to the floating-point implementation.     

A STACKELBERG DIFFERENTIAL GAMES SOLUTION TO A PROBLEM OF OPTIMAL INTERTEMPORAL INVESTMENT AND TAX RATE DESIGN

Using the Stackelberg differential games(SDG) theory, we quantitatively study a problem of optimal intertemporal investment and tax rate design. Under some appropriate assumptions, the open-loop Stackelberg equilibrium solutions are obtained. Equilibrium solutions show that: 1. The optimal strategies derived from differential game and unilateral optimal control approaches are different; 2. It is not always the best strategy for the government to use a constant tax rate over the whole time period; 3. The admissible size of tax rate adjustment may have great effect on the government's optimal strategy; 4. SDG approach has no significant effect on the firm's optimal investment strategy.     

Liouvillian Solutions of Algebraic Ordinary Differential Equations of Order One of Genus Zero

This paper considers the class of autonomous algebraic ordinary differential equations(AODEs) of order one, and studies their Liouvillian general solutions. In particular, let F(y, w) = 0 be a rational algebraic curve over C. The authors give necessary and sufficient conditions for the autonomous first-order AODE F(y, y′) = 0 to have a Liouvillian solution over C. Moreover, the authors show that a Liouvillian solution α of this equation is either an algebraic function over C(x) or an algebraic f...     

MERGING OPTIMALITY CONDITIONS WITH GENETIC ALGORITHM OPERATORS TO SOLVE SINGLE MACHINE TOTAL WEIGHTED TARDINESS PROBLEM

1.Introduction Production sequencing and scheduling is one of the most important activities in production planning and control.Sequencing is defined as the order in which the jobs are processed through the machines.The allocation of machines over time to process a collection of jobs is defined as Scheduling.Several methods have been developed to solve the scheduling problem which can be classified as follows:1)efficient optimal methods,2)implicit and explicit,or complete enumerative methods,a…     

Exponential distribution-based genetic algorithm for solving mixed-integer bilevel programming problems

Two classes of mixed-integer nonlinear bilevel programming problems are discussed. One is that the follower＇s functions are separable with respect to the follower＇s variables, and the other is that the follower＇s functions are convex if the follower＇s variables are not restricted to integers. A genetic algorithm based on an exponential distribution is proposed for the aforementioned problems. First, for each fixed leader＇s variable x, it is proved that the optimal solution y of the follower＇s mixed-integer programming can be obtained by solving associated relaxed problems, and according to the convexity of the functions involved, a simplified branch and bound approach is given to solve the follower＇s programming for the second class of problems. Furthermore, based on an exponential distribution with a parameter λ, a new crossover operator is designed in which the best individuals are used to generate better offspring of crossover. The simulation results illustrate that the proposed algorithm is efficient and robust.     

OPTIMAL RATE OF CONVERGENCE FOR ESTIMATORS IN THE NONPARAMETRIC MEDIAN MODEL

Let(Y,X)be a random vector with its value in R~1×R~d,d(?)1.Let(?)be the collection of real valued functions θ(x) on R~d which is p times differ-entiable at x=0 and p-1 times differentiable on an open neighborhood U ofthe origin of R~d.The conditional distribution of Y is assumed to be of the formof f(y|x,θ(x))dy where θ(x)∈(?)is called the parameter of the family.(Y,X)iscalled a nonparametric median model if furthermore the conditional median of Ygiven X=x is θ(x).In this paper,the optimal rate of convergence for estimatorsof T(θ)=θ(0)is discussed.Under certain conditions,it is proved that for thenonparametric median model the optimal rate of convergence is r=p/(2p+d).A sequence of estimators,which is asymptotically normal with the optimal rate ofcovergence,is constructed.     

A PRIOR ESTIMATE AND EXISTENCE OF POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATION WITH THE THIRD BOUNDARY VALUE PROBLEM

1 IntroductionLet n be a bounded open domain with boundary aa, n c R", N 3 2. We consider a priorestimate and ekistence of positive solution of the fOllowing equationsWhere operatorL conforms to the Sum Ru1es, that is: for a repeated i or j, it mean to sum them up from 1 toN, ai;(x), ai(x), a(x) are bounded measurable in a. FOr operator L, we alWays assume: thereealst positive constants po and U1) such thatf(u) is a nonlinear function which grows in the form of Up with p 2 1, n is the out…     

A proximal point algorithm for a system of generalized mixed variational inequalities

This paper introduces and considers a new system of generalized mixed variational inequalities in a Hilbert space,which includes many new and known systems of variational inequalities and generalized variational inequalities as special cases.By using the two concepts of 77-subdifferential andη-proximal mappings of a proper function,the authors try to demonstrate that the system of generalized mixed variational inequalities is equivalence with a fixed point problem.By applying the equivalence, a new and innovativeη-proximal point algorithm for finding approximate solutions of the system of generalized mixed variational inequalities will be suggested and analyzed.The authors also study the convergence analysis of the new iterative method under much weaker conditions.The results can be viewed as a refinement and improvement of the previously known results for variational inequalities.     

System deployment optimization in architecture design

Optimization of architecture design has recently drawn research interest. System deployment optimization （SDO） refers to the process of optimizing systems that are being deployed to activi- ties. This paper first formulates a mathematical model to theorize and operationalize the SDO problem and then identifies optimal so- lutions to solve the SDO problem. In the solutions, the success rate of the combat task is maximized, whereas the execution time of the task and the cost of changes in the system structure are mini- mized. The presented optimized algorithm generates an optimal solution without the need to check the entire search space. A novel method is finally proposed based on the combination of heuristic method and genetic algorithm （HGA）, as well as the combination of heuristic method and particle swarm optimization （HPSO）. Experi- ment results show that the HPSO method generates solutions faster than particle swarm optimization （PSO） and genetic algo- rithm （GA） in terms of execution time and performs more efficiently than the heuristic method in terms of determining the best solution.     

Multiobjective Decision Making: the Combined Approach

: In this paper, a new approach for generating all or partly efficient solutions called the Combined Approach is developed. The property of efficient solutions generated by the combined approach and its relationships with other four approaches: weighting approach, sequential approach, ε-constraint approach and hybrid approach, are discussed. Based on this combined approach, a decision-making support method called the Combined Decision-Making Method (CDMM) for multiobjective problems is developed, which is an interactive process with the decision maker. Only the aspiration levels, which reflect the decision maker's satisfying degrees for corresponding objectives, are needed to be supplied by the decision maker step by step as he will. This interactive way for objectives can easily be accepted. Finally, the application of the proposed decision making method in the resource allocation problem is discussed, and an example for the production decision analysis of the solar energy cells given.