A class of gap functions for quasi-variational inequality problems
Masao Fukushima
Journal of Industrial & Management Optimization 2007, 3(2): 165-171 doi: 10.3934/jimo.2007.3.165
We present a class of gap functions for the quasi-variational inequality problem (QVIP). We show the equivalence between the optimization reformulation with the gap function and the original QVIP. We also give conditions under which the gap function is continuous and directionally differentiable.
keywords: Quasi-variational inequality problem optimization reformulation gap function.
A class of stochastic mathematical programs with complementarity constraints: reformulations and algorithms
Gui-Hua Lin Masao Fukushima
Journal of Industrial & Management Optimization 2005, 1(1): 99-122 doi: 10.3934/jimo.2005.1.99
We consider a class of stochastic mathematical programs with equilibrium constraints (SMPECs), in which all decisions are required to be made here-and-now, before a random event is observed. We show that this kind of SMPEC plays a very important role in practice. In order to develop effective algorithms, we first give some reformulations of the SMPEC and then, based on these reformulations, we propose a smoothed penalty approach for solving the problem. A comprehensive convergence theory is also included.
keywords: stochastic mathematical program with equilibrium constraints hereand- now decision problem stationarity subdifferential convergence.
Robust portfolio selection with a combined WCVaR and factor model
Ke Ruan Masao Fukushima
Journal of Industrial & Management Optimization 2012, 8(2): 343-362 doi: 10.3934/jimo.2012.8.343
In this paper, a portfolio selection model with a combined Worst-Case Conditional Value-at-Risk (WCVaR) and Multi-Factor Model is proposed. It is shown that the probability distributions in the definition of WCVaR can be determined by specifying the mean vectors under the assumption of multivariate normal distribution with a fixed variance-covariance matrix. The WCVaR minimization problem is then reformulated as a linear programming problem. In our numerical experiments, to compare the proposed model with the traditional mean variance model, we solve the two models using the real market data and present the efficient frontiers to illustrate the difference. The comparison reveals that the WCVaR minimization model is more robust than the traditional one in a market recession period and it can be used in a long-term investment.
keywords: linear programming. multi-factor model Portfolio selection worst-case conditional value-at-risk
Global optimization via differential evolution with automatic termination
Bun Theang Ong Masao Fukushima
Numerical Algebra, Control & Optimization 2012, 2(1): 57-67 doi: 10.3934/naco.2012.2.57
Evolutionary Algorithms (EAs) provide a very powerful tool for solving optimization problems. In the last decades, numerous studies have been focusing on improving the performance of EAs. However, there is a lack of studies that tackle the question of the termination criteria. Indeed, EAs still need termination criteria prespecified by the user. In this paper, we propose to combine the Differential Evolution (DE) method with novel elements, i.e., the ``Gene Matrix'' (GM), the ``Space Decomposition'' (SD) and ``Space Rotation'' (SR) mechanisms, in order to equip DE with an automatic termination criterion without resort to predefined conditions. We name this algorithm ``Differential Evolution with Automatic Termination'' (DEAT). Numerical experiments using a test bed of widely used benchmark functions in 10, 50 and 100 dimensions show the effectiveness of the proposed method.
keywords: space decomposition. termination criteria Gene matrix Global optimization space rotation differential evolution
trust region method for nonsmooth convex optimization
Nobuko Sagara Masao Fukushima
Journal of Industrial & Management Optimization 2005, 1(2): 171-180 doi: 10.3934/jimo.2005.1.171
We propose an iterative method that solves a nonsmooth convex optimization problem by converting the original objective function to a once continuously differentiable function by way of Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. Under this setting, Fukushima and Qi (1996) and Rauf and Fukushima (2000) proposed a proximal Newton method and a proximal BFGS method, respectively, for nonsmooth convex optimization. While these methods employ a line search strategy to achieve global convergence, the method proposed in this paper uses a trust region strategy. We establish global and superlinear convergence of the method under appropriate assumptions.
keywords: strong convexity inexact function and gradient evaluations. Nonsmooth convex optimization BFGS method Moreau-Yosida regularization trust region method
An implicit programming approach for the road pricing problem with nonadditive route costs
Rhoda P. Agdeppa Nobuo Yamashita Masao Fukushima
Journal of Industrial & Management Optimization 2008, 4(1): 183-197 doi: 10.3934/jimo.2008.4.183
Road pricing is considered one of the effective means to reduce traffic congestion and environmental damage, and it has been introduced in major highways of most countries. The road pricing problem can be formulated as a mathematical program with equilibrium constraints (MPEC) and the resulting MPEC can be solved efficiently by the implicit programming approach if the user's route costs are additive. However, route costs are generally nonadditive in the real world. In this paper we consider road pricing on the traffic equilibrium problem with nonadditive route costs based on users' disutility functions. We then show that this formulation can be reformulated as a mathematical program with strictly monotone mixed complementarity problem (MCP). Since a strictly monotone MCP has a unique solution for each upper level variable, we can apply the implicit programming approach to solve the resulting reformulation. We establish the differentiability of the resulting implicit function. Numerical experiments using various disutility functions and sample networks are done, and the results show that the implicit programming approach is robust to find a solution of the road pricing problem.
keywords: mathematical program with equilibrium constraints nonadditive route cost Road pricing traffic equilibrium problem implicit programming method.

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