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.

. Motivated by block partitioned problems arising from group sparsity representation and generalized noncooperative potential games, this paper presents a basic decomposition method for a broad class of multiblock nonsmooth optimization problems subject to coupled linear constraints on the variables that may additionally be individually constrained. The objective of such an optimization problem is given by the sum of two nonseparable functions minus a sum of separable, pointwise maxima of finitely many convex differentiable functions. One of the former two nonseparable functions is of the class LC1 , i.e., differentiable with a Lipschitz gradient, while the other summand is multiconvex. The subtraction of the separable, pointwise maxima of convex functions induces a partial difference-of-convex (DC) structure in the overall objective; yet with all three terms together, the objective is nonsmooth and non-DC, but is blockwise directionally differentiable. By taking advantage of the (negative) pointwise maximum structure in the objective, the developed algorithm and its convergence result are aimed at the computation of a blockwise directional stationary solution, which arguably is the sharpest kind of stationary solutions for this class of nonsmooth problems. This aim is accomplished by combining the alternating direction method of multipliers (ADMM) with a semilinearized Gauss–Seidel scheme, resulting in a decomposition of the overall problem into subproblems each involving the individual blocks. To arrive at a stationary solution of the desired kind, our algorithm solves multiple convex subprograms at each iteration, one per convex function in each pointwise maximum. In order to lessen the potential computational burden in each iteration, a probabilistic version of the algorithm is presented and its almost sure convergence is established.

In this paper, we formulate a problem of simultaneous redundant via insertion and line end extension for via yield optimization. Our problem is more general than previous works in the sense that more than one type of line end extension is considered and the objective function to be optimized directly accounts for via yield. We present a zero-one integer linear program based approach, that is equipped with two speedup techniques, to solve the addressed problem optimally. In addition, we describe how to modify our approach to exactly solve a previous work. Extensive experimental results are shown to demonstrate the effectiveness and efficiency of our approaches.

This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and non-negative constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.

We propose a class of interior proximal-like algorithms for the second-order cone program, which is to minimize a closed proper convex function subject to general second-order cone constraints. The class of methods uses a distance measure generated by a twice continuously differentiable strictly convex function on (0,+\infty), and includes as a special case the entropy-like proximal algorithm [Eggermont, <i>Linear Algebra Appl.</i>, 130 (1990), pp. 2542], which was originally proposed for minimizing a convex function subject to nonnegative constraints. Particularly, we consider an approximate version of these methods, allowing the inexact solution of subproblems. Like the entropy-like proximal algorithm for convex programming with nonnegative constraints, we, under some mild assumptions, establish the global convergence expressed in terms of the objective values for the proposed algorithm, and we show that the