Inner and outer estimates for the solution sets and their asymptotic cones in vector optimization

We use asymptotic analysis to develop finer estimates for the efficient, weak efficient and proper efficient solution sets (and for their asymptotic cones) to convex/quasiconvex vector optimization problems. We also provide a new representation for the efficient solution set without any convexity assumption, and the estimates involve the minima of the linear scalarization of the original vector problem. Some new necessary conditions for a point to be efficient or weak efficient solution for general convex vector optimization problems, as well as for the nonconvex quadratic multiobjective optimization problem, are established.     

Search for Efficient Solutions of Multi-Criterion Problems by Target-Level Method

The target-level method is considered for solving continuous multi-criterion maximization problems. In the first step, the decision-maker specifies a target-level point (the desired criterion values); then in the set of vector evaluations we seek points that are closest to the target point in the Chebyshev metric. The vector evaluations obtained in this way are in general weakly efficient. To identify the efficient evaluations, the second step maximizes the sum of the criteria on the set generated in step 1. We prove the relationship between the evaluations and decisions obtained by the proposed procedure, on the one hand, and the efficient (weakly efficient) evaluations and decisions, on the other hand. If the Edgeworth–Pareto hull of the set of vector evaluations is convex, the set of efficient vector evaluations can be approximated by the proposed method.     

The Pascoletti–Serafini Scalarization Scheme and Linear Vector Optimization

The Pascoletti–Serafini scalarization scheme for general vector optimization problems is studied. It is specified to linear vector optimization to give minimal representation formulae for the weakly efficient solution set and the efficient solution set. Several facts on connectedness of the solution sets of Pascoletti–Serafini’s scalar auxiliary problems, both for linear vector optimization and for nonlinear vector optimization, are established.     

On possibility of obtaining linear accuracy evaluation of approximate solutions to inverse problems

We consider a concept of linear a priori estimate of the accuracy for approximate solutions to inverse problems with perturbed data. We establish that if the linear estimate is valid for a method of solving the inverse problem, then the inverse problem is well-posed according to Tikhonov. We also find conditions, which ensure the converse for the method of solving the inverse problem independent on the error levels of data. This method is well-known method of quasi-solutions by V. K. Ivanov. It provides for well-posed (according to Tikhonov) inverse problems the existence of linear estimates. If the error levels of data are known, a method of solving well-posed according to Tikhonov inverse problems is proposed. This method called the residual method on the correctness set (RMCS) ensures linear estimates for approximate solutions. We give an algorithm for finding linear estimates in the RMCS.     

Linear bilevel programs with multiple objectives at the upper level

Bilevel programming has been proposed for dealing with decision processes involving two decision makers with a hierarchical structure. They are characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. Focus of the paper is on general bilevel optimization problems with multiple objectives at the upper level of decision making. When all objective functions are linear and constraints at both levels define polyhedra, it is proved that the set of efficient solutions is non-empty. Taking into account the properties of the feasible region of the bilevel problem, some methods of computing efficient solutions are given based on both weighted sum scalarization and scalarization techniques. All the methods result in solving linear bilevel problems with a single objective function at each level.     

Efficient multicriterial optimization based on intensive reuse of search information

This paper proposes an efficient method for solving complex multicriterial optimization problems, for which the optimality criteria may be multiextremal and the calculations of the criteria values may be time-consuming. The approach involves reducing multicriterial problems to global optimization ones through minimax convolution of partial criteria, reducing dimensionality by using Peano curves and implementing efficient information-statistical methods for global optimization. To efficiently find the set of Pareto-optimal solutions, it is proposed to reuse all the search information obtained in the course of optimization. The results of computational experiments indicate that the proposed approach greatly reduces the computational complexity of solving multicriterial optimization problems.     

Locally extra-optimal regularizing algorithms and a posteriori estimates of the accuracy for ill-posed problems with discontinuous solutions

Local a posteriori estimates of the accuracy of approximate solutions to ill-posed inverse problems with discontinuous solutions from the classes of functions of several variables with bounded variations of the Hardy or Giusti type are studied. Unlike global estimates (in the norm), local estimates of accuracy are carried out using certain linear estimation functionals (e.g., using the mean value of the solution on a given fragment of its support). The concept of a locally extra-optimal regularizing algorithm for solving ill-posed inverse problems, which has an optimal in order local a posteriori estimate, was introduced. A method for calculating local a posteriori estimates of accuracy with the use of some distinguished classes of linear functionals for the problems with discontinuous solutions is proposed. For linear inverse problems, the method is bases on solving specialized convex optimization problems. Examples of locally extra-optimal regularizing algorithms and results of numerical experiments on a posteriori estimation of the accuracy of solutions for different linear estimation functionals are presented.     

求解分布控制问题的预处理最小残量方法(英文)

通过分析Bai(Bai Z Z.Block preconditioners for elliptic PDE-constrained optimization problems.Computing,2011,91:379-395)给出的离散分布控制问题的块反对角预处理线性系统,提出了该问题的一个等价线性系统,并且运用带有预处理子的最小残量方法对该系统进行求解.理论分析和数值实验结果表明,所提出的预处理最小残量方法对于求解该类椭圆型偏微分方程约束最优分布控制问题非常有效,尤其当正则参数适当小的时候.     

On minkowski metric and weighted tchebyshev norm in vector optimization ∗

The typical approach in solving vector optimization problems is to scalarize the vector cost function into a single cost function by means of some utility or value function. A very large class of utility function is given by the Minkowski’s metric proposed by Charnes and Cooper in the context of goal programming. This includes the special case of linear scalarization and the weighted Tchebyshev norm. We shall furnish a rigorous justification that there is no equivalent relationship between the general vector optimization problem and scalarized optimization problems using any Minkowski’s metric utility function. Furthermore, we also show that the weighted Tchebyshev norm is, in some sense, the best amongst the class of Minkowski’s metric utility functions since it is the only scalarization method which yields an equivalence relation between the weak vector optimization problem and a set of scalar optimization problems, without any convexity assumption     

An interactive algorithm for nonlinear vector optimization

In this paper an interactive algorithm for nonlinear vector optimization problems is presented. This algorithm decides, after solving only two optimization problems, whether or not there are efficient points in the feasible set. In the latter case, an efficient point depending on parameters is automatically computed, and (which is much more important) efficient points for each parameter can be calculated by this procedure.     

Connectedness of efficient points in convex and convex transformable vector optimization

Given a convex vector optimization problem with respect to a closed ordering cone, we show the connectedness of the efficient and properly efficient sets. The Arrow–Barankin–Blackwell theorem is generalized to nonconvex vector optimization problems, and the connectedness results are extended to convex transformable vector optimization problems. In particular, we show the connectedness of the efficient set if the target function f is continuously transformable, and of the properly efficient set if f is differentiably transformable. Moreover, we show the connectedness of the efficient and properly efficient sets for quadratic quasiconvex multicriteria optimization problems.     

Newton-like methods for solving vector optimization problems

In the context of Euclidean spaces, we present an extension of the Newton-like method for solving vector optimization problems, with respect to the partial orders induced by a pointed, closed and convex cone with a nonempty interior. We study both exact and inexact versions of the Newton-like method. Under reasonable hypotheses, we prove stationarity of accumulation points of the sequences produced by Newton-like methods. Moreover, assuming strict cone-convexity of the objective map to the vector optimization problem, we establish convergence of the sequences to an efficient point whenever the initial point is in a compact level set.     

Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone

We define weakly minimal elements of a set with respect to a convex cone by means of the quasi-interior of the cone and characterize them via linear scalarization, generalizing the classical weakly minimal elements from the literature. Then we attach to a general vector optimization problem, a dual vector optimization problem with respect to (generalized) weakly efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem, we derive vector dual problems with respect to weakly efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.     

Composite Nonsmooth Optimization Using Approximate Generalized Gradient Vectors

A new approximation method is presented for directly minimizing a composite nonsmooth function that is locally Lipschitzian. This method approximates only the generalized gradient vector, enabling us to use directly well-developed smooth optimization algorithms for solving composite nonsmooth optimization problems. This generalized gradient vector is approximated on each design variable coordinate by using only the active components of the subgradient vectors; then, its usability is validated numerically by the Pareto optimum concept. In order to show the performance of the proposed method, we solve four academic composite nonsmooth optimization problems and two dynamic response optimization problems with multicriteria. Specifically, the optimization results of the two dynamic response optimization problems are compared with those obtained by three typical multicriteria optimization strategies such as the weighting method, distance method, and min–max method, which introduces an artificial design variable in order to replace the max-value cost function with additional inequality constraints. The comparisons show that the proposed approximation method gives more accurate and efficient results than the other methods.     

Criteria and dimension reduction of linear multiple criteria optimization problems

Two of the main approaches in multiple criteria optimization are optimization over the efficient set and utility function program. These are nonconvex optimization problems in which local optima can be different from global optima. Existing global optimization methods for solving such problems can only work well for problems of moderate dimensions. In this article, we propose some ways to reduce the number of criteria and the dimension of a linear multiple criteria optimization problem. By the concept of so-called representative and extreme criteria, which is motivated by the concept of redundant (or nonessential) objective functions of Gal and Leberling, we can reduce the number of criteria without altering the set of efficient solutions. Furthermore, by using linear independent criteria, the linear multiple criteria optimization problem under consideration can be transformed into an equivalent linear multiple criteria optimization problem in the space of linear independent criteria. This equivalence is understood in a sense that efficient solutions of each problem can be derived from efficient solutions of the other by some affine transformation. As a result, such criteria and dimension reduction techniques could help to increase the efficiency of existing algorithms and to develop new methods for handling global optimization problems arisen from multiple objective optimization.     

A surrogate based multi-fidelity approach for robust design optimization

Robust design optimization (RDO) is a field of optimization in which certain measure of robustness is sought against uncertainty. Unlike conventional optimization, the number of function evaluations in RDO is significantly more which often renders it time consuming and computationally cumbersome. This paper presents two new methods for solving the RDO problems. The proposed methods couple differential evolution algorithm (DEA) with polynomial correlated function expansion (PCFE). While DEA is utilized for solving the optimization problem, PCFE is utilized for calculating the statistical moments. Three examples have been presented to illustrate the performance of the proposed approaches. Results obtained indicate that the proposed approaches provide accurate and computationally efficient estimates of the RDO problems. Moreover, the proposed approaches outperforms popular RDO techniques such as tensor product quadrature, Taylor’s series and Kriging. Finally, the proposed approaches have been utilized for robust hydroelectric flow optimization, demonstrating its capability in solving large scale problems.     

Optimal estimation of sensor biases for asynchronous multi-sensor data fusion

An important step in a multi-sensor surveillance system is to estimate sensor biases from their noisy asynchronous measurements. This estimation problem is computationally challenging due to the highly nonlinear transformation between the global and local coordinate systems as well as the measurement asynchrony from different sensors. In this paper, we propose a novel nonlinear least squares formulation for the problem by assuming the existence of a reference target moving with an (unknown) constant velocity. We also propose an efficient block coordinate decent (BCD) optimization algorithm, with a judicious initialization, to solve the problem. The proposed BCD algorithm alternately updates the range and azimuth bias estimates by solving linear least squares problems and semidefinite programs. In the absence of measurement noise, the proposed algorithm is guaranteed to find the global solution of the problem and the true biases. Simulation results show that the proposed algorithm significantly outperforms the existing approaches in terms of the root mean square error.     

Improvement sets and vector optimization

In this paper we focus on minimal points in linear spaces and minimal solutions of vector optimization problems, where the preference relation is defined via an improvement set E. To be precise, we extend the notion of E-optimal point due to Chicco et al. in [4] to a general (non-necessarily Pareto) quasi ordered linear space and we study its properties. In particular, we relate the notion of improvement set with other similar concepts of the literature and we characterize it by means of sublevel sets of scalar functions. Moreover, we obtain necessary and sufficient conditions for E-optimal solutions of vector optimization problems through scalarization processes by assuming convexity assumptions and also in the general (nonconvex) case. By applying the obtained results to certain improvement sets we generalize well-known results of the literature referred to efficient, weak efficient and approximate efficient solutions of vector optimization problems.     

Efficient solutions in generalized linear vector optimization

This paper establishes several new facts on generalized polyhedral convex sets and shows how they can be used in vector optimization. Among other things, a scalarization formula for the efficient solution sets of generalized linear vector optimization problems is obtained. We also prove that the efficient solution set of a generalized linear vector optimization problem in a locally convex Hausdorff topological vector space is the union of finitely many generalized polyhedral convex sets and it is connected by line segments.     

Semidefinite relaxations for semi-infinite polynomial programming

This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation methods. We first recall two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.