It is well known that complementarity functions play an important role in dealing with complementarity problems. In this paper, we propose a few new classes of complementarity functions for nonlinear complementarity problems and second-order cone complementarity problems. The constructions of such new complementarity functions are based on discrete generalization which is a novel idea in contrast to the continuous generalization of FischerBurmeister function. Surprisingly, these new families of complementarity functions possess continuous differentiability even though they are discrete-oriented extensions. This feature enables that some methods like derivative-free algorithm can be employed directly for solving nonlinear complementarity problems and second-order cone complementarity problems. This is a new discovery to the literature and we believe that such new complementarity functions

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of O(k^{-1/(2^{d+1}-2)}), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of O(1/k^{0.5}).

In Nash bargaining problem, due to fairness concerns of players, instead of maximizing the sum of utilities of all players, an implementable solution should satisfy some axioms or characterizations. Such a solution can result in the so-called price of fairness, because of the reduction in the sum of utilities of all players. An important issue is to quantify the system efficiency loss under axiomatic solutions through the price of fairness. Based on Perles–Maschler solution of two-player Nash bargaining problem, this paper deals with the extended Perles–Maschler solution of multi-player Nash bargaining problem.We give lower bounds of threemeasures of the system efficiency for this solution, and show that the lower bounds are asymptotically tight.

Many iterative methods in optimization are fixed-point iterations with averaged operators. As such methods converge at an O(1/k) rate with the constant determined by the averagedness coefficient, establishing small averagedness coefficients for operators is of broad interest. In this paper, we show that the averagedness coefficients of the composition of averaged operators by Ogura and Yamada (Numer Func Anal Opt 32(1–2):113–137, 2002) and the threeoperator splitting by Davis and Yin (Set Valued Var Anal 25(4):829–858, 2017) are tight. The analysis relies on the scaled relative graph, a geometric tool recently proposed by Ryu, Hannah, and Yin (arXiv:1902.09788, 2019).

No abstract uploaded!