The relation between Pareto, Slater, Geoffrion, and potential optimality is investigated for
basic classes of value functions in multicriterial optimization problems.
Regularized equations for binary mixtures of viscous compressible gases (in the absence of chemical reactions) are considered. Two new simpler systems of equations are constructed for the case of a homogeneous mixture, when the velocities and temperatures of the components coincide. In the case of moderately rarefied gases, such a system is obtained by aggregating previously derived general equations for binary mixtures of polyatomic gases. In the case of relatively dense gases, the regularizing terms in these equations are subjected to a further substantial modification. For both cases, balance equations for the total mass, kinetic, and internal energy and new balance equations for total entropy are derived from the constructed equations; additionally, the entropy production is proved to be nonnegative. As an example of successful use of the new equations, the two-dimensional Rayleigh–Taylor instability of relatively dense gas mixtures is numerically simulated in a wide range of Atwood numbers.
An approach to sensitivity (stability) analysis of nondominated alternatives to changes in the bounds of intervals of value tradeoffs, where the alternatives are selected based on interval data of criteria tradeoffs is proposed. Methods of computations for the analysis of sensitivity of individual nondominated alternatives and the set of such alternatives as a whole are developed.
For the quasi-gasdynamic system of equations, there holds the law of nondecreasing entropy. Difference methods based on this system have been successfully used in numerous applications and test gasdynamic computations. In theoretical terms, however, for standard spatial discretizations of this system, the nondecreasing entropy law does not hold exactly even in the one-dimensional case because of the mesh imbalance terms. For the quasi-gasdynamic equations, a new conservative spatial discretization is proposed for which the entropy balance equation has an appropriate form and the entropy production is guaranteed to be nonnegative (which also holds in the presence of body forces and heat sources). An important element of this discretization is that it makes use of nonstandard space-averaging techniques, including a nonlinear "logarithmic" averaging of the density and internal energy. The results hold on arbitrary nonuniform meshes
A universal method of searching for usual and stochastic equilibria in congestion population games is proposed. The Beckmann and stable dynamics models of an equilibrium flow distribution over paths are considered. A search for Nash(–Wardrop) stochastic equilibria leads to entropy-regularized convex optimization problems. Efficient solutions of such problems, more exactly, of their duals are sought by applying a recently proposed universal primal-dual gradient method, which is optimally and adaptively tuned to the smoothness of the problem under study. © 2019, Pleiades Publishing, Ltd.
An exact computational method is proposed for the preferability comparison of various solution variants in multicriteria problems withimportanceordered criteria using a common scale along which the growth of preferences slows down.
Exact efficient numerical methods are proposed for solving bilinear optimization problems that arise when various solution variants are compared based on their preferability using an additive value function in the case of interval estimates of the degrees of superiority of certain criteria over others and in the case of interval restrictions on the growth of preferences along the criteria range.