Choquet integral for criteria aggregation in the flexible job-shop scheduling problems

Most complex scheduling problems are combinatorial problems and difficult to solve. That is why, several methods focus on the optimization according to a single criterion such as makespan, workloads of machines, waiting times, etc. In this paper, the Choquet integral is introduced as a general tool for dealing with multiple criteria decision making and used in optimization flexible job-shop scheduling problems. The considered optimization problem is based of the Genetic Algorithm (GA) used as objective function the Choquet integral for criteria aggregation. Then lower bounds are defined for each criterion. Presented examples illustrate theoretical considerations and show the efficiency of the proposed approach.     

On stochastic linear programming problems with the quantile criterion

We study qualitative properties of a stochastic linear programming problem with quantile criterion for a wide class of distributions. We show convexity conditions for the criterion function with respect to the strategy, and continuity conditions with respect to the strategy and reliability level. We give sufficient conditions for the existence of a solution. We present a new algorithm for finding a guaranteeing solution of the problem, i.e., an admissible solution for which the quantile criterion function's value turns out to be close to optimal.     

Stochastic quasigradient algorithm to minimize the function of integral quantile

Proposed was a stochastic quasigradient algorithm to minimize the integral quantile function (CVaR criterion) with the use of the order statistics. Convergence of the algorithm with the probability one was proved, and an example was considered.     

Bilevel stochastic linear programming problems with quantile criterion

We propose a setting for a bilevel stochastic linear programming problem with quantile criterion. We study continuity properties of the criterial function and prove the existence theorem for a solution. We propose a deterministic equivalent of the problem for the case of a scalar random parameter. We show an equivalent problem in the form of a two-stage stochastic programming problem with equilibrium constraints and quantile criterion. For the case of a discrete distribution of random parameters, the problem reduces to a mixed linear programming problem. We show results of numerical experiments.     

Equivalence of quadratic performance criteria†

Two quadratic cost functionals are defined to be equivalent, if they generate the same optimal control law. Necessary and sufficient conditions are derived in terms of the system parameters and the quadratic weighting matrices for two cost functionals to be equivalent.     

Equivalence problems in structural identifiability

This paper presents a reformulation of the identifiability problem, on the basis of equivalence between a model and the process it represents. Following a brief review of the major definitions in the literature, the concept of “near-equivalence” is introduced for both theoretical and real world identification problems. This concept is illustrated with a general block diagram and two examples. It is concluded that the theory of identifiability needs to be reformulated if it is be broadly applicable to the real world situations.     

Viscosity solutions and optimal control problems with integral constraints

We discuss optimal control problems with integral state-control constraints. We rewrite the problem in an equivalent form as an optimal control problem with state constraints for an extended system, and prove that the value function, although possibly discontinuous, is the unique viscosity solution of the constrained boundary value problem for the corresponding Hamilton–Jacobi equation. The state constraint is the epigraph of the minimal solution of a second Hamilton–Jacobi equation. Our framework applies, for instance, to systems with design uncertainties.     

Equivalence condition of optimal control problems†

Two optimal control problems which have the same optimal feedback control laws and identical minimum costs are said to be equivalent. The conditions for the equivalence of two optimal control problems have been presented by other authors, but it seems that the equivalence conditions given by other authors are not complete. In this paper, a necessary and sufficient condition for the equivalence classes is derived.     

Deterministic equivalents for the problems of stochastic programming with probabilistic criteria

Consideration was given to the problems of stochastic programming with probabilistic and quantile criteria. For some special cases, deterministic equivalents of the original stochastic problems were constructed. The convex properties of the probabilistic and quantile criteria were established for the problems under study.     

Equivalence of regularizations in abstract problems with different classes of admissible controls

Conclusion Theorems 4.1, 4.2 and 6.1, 6.2 respectively admit a natural specialization for the problem of constructing the reachability region of the linear controlled system from Sec. 1. Informally, this specialization has the following form. If the vector functions b(·) and S(·) from Sec. 1 are “not too discontinuous” (admit a uniform approximation by piecewise-constant and right-continuous maps on [t₀, v_oD, then, given a common resource constraintc, the controlled analogues of reachability regions are identical for the class of controls with an integral constraint (on the total pulse) and the class of “pure pulse” shock controls, whereas the “ordinary” reachability regions corresponding to unperturbed conditions (see, e.g., they− constraint in (1.2)) may be different. This is illustrated by the examples of Sees. 1, 5. The regularized version of the problem of constructing the reachability region of a linear system is thus insensitive to a change of the class of admissible controls. The study was supported by the Russian Foundation for Basic Research (94-01-00350). Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 3–17, May–June, 1998.     

Mirror scheduling problems with early work and late work criteria

Journal of Scheduling - We give a definition of a class of mirror scheduling problems, review existing representatives of this class and demonstrate that an identical parallel machine scheduling...     

Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations

Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper. Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter. It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.     

Reduction of some linear optimal control problems with integral constraints

We consider the relationship between the continuous and the impulse control forms of the problem of sequential optimization of a finite system of deviations. Dual mathematical programming constructs are considered. Equivalence conditions by the end result are established for control procedures using ordinary and simple generalized functions.Translated from Kibernetika, No. 4, pp. 59–64, July–August, 1990.     

New stability criteria of singular systems with time-varying delay via free-matrix-based integral inequality

This paper concerns the stability problem of singular systems with time-varying delay. First, the singular system with time-varying delay is transformed into the neutral system with time-varying delay. Second, a more proper Lyapunov–Krasovskii functional (LKF) is constructed by adding some integral terms to quadratic forms. Then, to obtain less conservative conditions, the free-matrix-based integral inequality is adopted to estimate the derivative of LKF. As a result, some delay-dependent stability criteria are given in terms of linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness and superiority of the proposed method.