Efim A. Galperin

The cubic algorithm

Abstract A nongradient algorithm for nonlinear nonconvex Lipschitzian optimization problems is proposed.

Nonscalarized multiobjective global optimization

A new approach to multiobjective optimization is presented which is made possible due to our ability to obtain full global optimal solutions. A distinctive feature of this approach is that a vector cost function is nonscalarized. The method provides a means for the solution of vector optimization problems with nonreconcilable objectives.

Box-triangular multiobjective linear programs for resource allocation with application to load management and energy market problems

Models for multicriteria resource allocation are constructed with the specific box-triangular structure of a feasible region. The method of balance set equations is extended for the satisfaction level representation of the cost function space including the case of linearly dependent cost functions. On this basis, different goal criteria on the balance set are investigated for linear cases. Procedures for determining the balance set and finding goal-optimal Pareto solutions are illustrated on examples. The results of the paper are of universal character and can find wide applications in allocating diverse types of resources on the multiobjective basis in planning and control of complex systems including load management and energy market problems.

Retrieval and use of the balance set in multiobjective global optimization

Abstract It is shown, on examples, how to compute the balance set and the balance number in Vector Optimization Problems (VOPs) of different nature. New developments are presented concerning possible interrelation between the balance set and the balance number, a new notion of the projection of the balance set onto the parameter space, new approaches for solving VOPs with unbounded objective functions, and some approximation techniques in determining the balance set.

Pareto analysis vis-à-vis balance space approach in multiobjective global optimization

There is much controversy about the balance space approach, introduced first in Ref. 1, pp. 138–140, with the consideration of the balance number and balance vectors, and then further developed in Ref. 2, with the consideration of balance points and balance sets. There were attempts to identify the balance space approach with some other methods of multiobjective optimization, notably the method proposed in Ref. 3 and most recently Pareto analysis, as presented in Ref. 4. In this paper, we compare Pareto analysis with the balance space approach on several examples to demonstrate the interrelation and the differences of the two methods. As a byproduct, it is shown that, in some cases, the entire Pareto sets, proper and adjoint, can be determined very simply, without any special investigation of the (nonscalarized, nonconvex) multiobjective global optimization problem. The method of parameter introduction is presented in application to determining the Pareto sets and balance set. The use of computer graphics software complemented with the Gauss–Jordan matrix reduction algorithm is proposed for a class of otherwise intractable problems with nonconvex constraint sets.

Precision, complexity, and computational schemes of the cubic algorithm

The cubic algorithm (Ref. 1) is a nongradient method for the solution of multi-extremal, nonconvex Lipschitzian optimization problems. The precision and complexity of this algorithm are studied, and improved computational schemes are proposed.

Min-max formulation of the balance number in multiobjective global optimization

Abstract The notion of the balance number introduced by Galperin through a certain set contraction procedure for nonscalarized multiobjective global optimization is represented via a min-max operation on the data of the problem. This representation yields a different computational procedure for the calculation of the balance number and allows us to generalize the approach for problems with countably many performance criteria. Comparisons with Pareto optimality and compromise solutions are discussed and illustrated by examples. It is demonstrated that l p -norm scalarizations (1 ≤ p

Duality of nonscalarized multiobjective linear programs: dual balance, level sets, and dual clusters of optimal vectors

A new concept of duality is proposed for multiobjective linear programs. It is based on a set expansion process for the computation of optimal solutions without scalarization. The duality gap qualifications are investigated; the primal–dual balance set and level set equations are derived. It is demonstrated that the nonscalarized dual problem presents a cluster of optimal dual vectors that corresponds to a unique optimal primal vector. Comparisons are made with linear utility, minmax and minmin scalarizations. Connections to Pareto optimality are studied and relations to sensitivity and parametric programming are discussed. The ideas are illustrated by examples.

Solution and control of PDE via global optimization methods

Abstract Based on the concept of η-equivalent solutions (not to be confused with approximations to the exact solution), a new consideration is given to ill-posed [1,2] and overdetermined PDE problems and to problems with nonexistent solutions [3]. Then a new method based on full global optimization techniques is developed for solution and control of processes described by partial differential equations. The ideas are illustrated by examples, and a case study is presented in comparison with the quasi-reversibility method [4].

Nonlinear observation via global optimization methods: measure theory approach

Nonlinear observation methods developed by Galperin (Refs. 1 and 2) and global optimization methods developed by Zheng (Refs. 3 and 4) are coupled to obtain effective procedures for solution of nonlinear observation and identification problems.