Invited Session Thu.2.H 0107

Thursday, 13:15 - 14:45 h, Room: H 0107

Cluster 16: Nonlinear programming [...]

Algorithms and applications I


Chair: Ya-xiang Yuan



Thursday, 13:15 - 13:40 h, Room: H 0107, Talk 1

Coralia Cartis
On the evaluation complexity of constrained nonlinear programming

Coauthors: Nicholas I.m. Gould, Philippe L. Toint


We present a short-step target-following algorithm for smooth and nonconvexly constrained
programming problems that relies upon approximate first-order minimization of a nonsmooth
composite merit function and that takes at most O-2) problem-evaluations to generate an approximate KKT point or an infeasible point of the feasibility measure. This bound has the same order as that for steepest-descent methods applied to unconstrained problems. Furthermore, complexity bounds of (optimal) order ε-3/2 are obtained if cubic regularization steps for a smooth least-squares merit function are employed in a similar target-following algorithmic framework, provided higher accuracy is required for primal than for dual feasibility.



Thursday, 13:45 - 14:10 h, Room: H 0107, Talk 2

Xiao Wang
An augmented Lagrangian trust region method for nonlinear programming

Coauthor: Ya-xiang Yuan


We present a new trust region method for solving
equality constrained optimization problems, which is motivated by the famous augmented Lagrangian function. Different from the standard augmented Lagrangian method where the augmented Lagrangian function is minimized at each iteration, the new method, for fixed Lagrange multiplier and penalty parameter, tries to minimize an approximation model to the augmented Lagrangian function in a trust region to generate next iterate. Besides, new update strategies for Lagrange multipliers and penalty parameters are proposed. Global convergence of the new algorithm is proved in this paper. Moreover, we analyze the behavior of penalty parameters and figure out in which case when they are bounded. At last, we do some numerical experiments on the equality constrained problems from CUTEr collection. We also consider extending the idea to general constrained optimization. Some numerical results are reported too.



Thursday, 14:15 - 14:40 h, Room: H 0107, Talk 3

Zhijun Wu
Computation of optimal strategies for evolutionary games

Coauthors: Yiping Hao, Wen Zhou


Biological species (viruses, bacteria, parasites, insects, plants, or animals) replicate, mutate, compete, adapt, and evolve. In evolutionary game theory, such a process is modeled as a so-called evolutionary game. We describe the Nash equilibrium problem for an evolutionary game and discuss its computational complexity. We discuss the necessary and sufficient conditions for the equilibrium states, and derive the methods for the computation of the optimal strategies, including a specialized Snow-Shapley algorithm, a specialized Lemke-Howson algorithm, and an algorithm based on the solution of a complementarity problem on a simplex. Computational results are presented. Theoretical difficulties and computational challenges are highlighted.


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