JIMO
Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities
Gang Qian Deren Han Lingling Xu Hai Yang
Journal of Industrial & Management Optimization 2013, 9(1): 255-274 doi: 10.3934/jimo.2013.9.255
In the last decade, as calibrations of the classical traffic equilibrium problems, various models of traffic equilibrium problems with nonadditive route costs have been proposed. For solving such models, this paper develops a self-adaptive projection-auxiliary problem method for monotone variational inequality (VI) problems. It first converts the original problem where the feasible set is the intersection of a linear manifold and a simple set to an augmented VI with simple set, which makes the projection easy to implement. The self-adaptive strategy avoids the difficult task of choosing `suitable' parameters, and leads to fast convergence. Under suitable conditions, we prove the global convergence of the method. Some preliminary computational results are presented to illustrate the ability and efficiency of the method.
keywords: auxiliary problem principle self-adaptive strategy. projection method Traffic equilibrium and nonadditive cost variational inequality problem
JIMO
A proximal alternating direction method for multi-block coupled convex optimization
Foxiang Liu Lingling Xu Yuehong Sun Deren Han
Journal of Industrial & Management Optimization 2018, 13(5): 1-15 doi: 10.3934/jimo.2018067

In this paper, we propose a proximal alternating direction method (PADM) for solving the convex optimization problems with linear constraints whose objective function is the sum of multi-block separable functions and a coupled quadratic function. The algorithm generates the iterate via a simple correction step, where the descent direction is based on the PADM. We prove the convergence of the generated sequence under some mild assumptions. Finally, some familiar numerical results are reported for the new algorithm.

keywords: Convex optimization correction step multi-block proximal alternating direction method coupled quadratic function
JIMO
A new parallel splitting descent method for structured variational inequalities
Kai Wang Lingling Xu Deren Han
Journal of Industrial & Management Optimization 2014, 10(2): 461-476 doi: 10.3934/jimo.2014.10.461
In this paper, we propose a new parallel splitting descent method for solving a class of variational inequalities with separable structure. The new method can be applied to solve convex optimization problems in which the objective function is separable with three operators and the constraint is linear. In the framework of the new algorithm, we adopt a new descent strategy by combining two descent directions and resolve the descent direction which is different from the methods in He (Comput. Optim. Appl., 2009, 42: 195-212.) and Wang et al. (submitted to J. Optimiz. Theory App.). Theoretically, we establish the global convergence of the new method under mild assumptions. In addition, we apply the new method to solve problems in management science and traffic equilibrium problem. Numerical results indicate that the new method is efficient and reliable.
keywords: variational inequalities Alternating direction method parallel computing augmented Lagrangian method separable structure.

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