Saturday, 14:20 - 16:00 - Room 146

*Stream: Conic optimization and semi-definite programming*

Chair:

- Plea for a semidefinite optimization solver in complex numbers

Numerical optimization in complex numbers has drawn much less attention than in real numbers. A widespread opinion is that, since a complex number is a pair of real numbers, the best strategy to solve a complex optimization problem is to transform it into real numbers and to solve the latter by a real number solver. This talk defends another point of view and presents arguments to convince the audience that skipping the transformation phase and using a complex number algorithm can be much more efficient. A speedup of two is not uncommon for interior point methods in complex SDP optimization.

- On Digital Filter Design with Semidefinite Programming

We shall consider an efficient digital filter design by means of semidefinite programming (SDP) methods. Formulations of some filter design goals as optimization problems are presented. This novel approach allows for extra requirements about the filter, such as a constant and small group delay, to be fulfilled. Numerical examples prove that employing state-of-the-art methods and solvers create new possibilities for easier design of higher-quality digital filters.

- Solutions of Complementarity Problems Monotone with Respect to Parameters

In many applied problems (such as, e.g., elastohydrodynamic lubrication problem, some economic equilibrium problems, etc., one of the important question is if certain complementarity problem's solution is monotone with respect to parameters. Our paper investigates this question and provides several sufficient conditions that guarantee such a monotonicity of the solutions to linear and nonlinear complementarity problems with parameters. In the majority of cases, it is required that the principal mapping of the complementarity problem be monotone by decision variables and, vice versa, antitone.

- Departure time choice equilibrium with heterogeneous commuters in corridor type traffic network

The purpose of this study is to model and analyze the departure time choice equilibrium with heterogeneous commuters in corridor type traffic network. We first formulate the dynamic user equilibrium (DUE) problem as a linear complementarity problem, and then study the existence and the uniqueness of the solution in an analytic or experimental manner. We also prove that the equilibrium solution is separated into each user group. We also make a number of numerical experiments to characterize the solution. Then we observe that the flow can be different essentially from that of homogeneous one.

- D-gap functions and optimization techniques for computing equilibria

A descent type algorithm for solving equilibrium problems with differentiable bifunctions is provided relying on a suitable family of D-gap functions. Its convergence is proved under assumptions not guaranteeing the equivalence between the stationary points of any D-gap function and the solutions of the equilibrium problem. Unlikely other algorithms, it does not require to set parameters according to thresholds depending on the equilibrium bifunction. Some numerical comparisons with these other algorithms are drawn relying on some extensive tests on the so-called linear equilibrium problems.