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Journal of Industrial & Management Optimization

2013 , Volume 9 , Issue 3

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Control parametrization and finite element method for controlling multi-species reactive transport in a circular pool
Heung Wing Joseph Lee, Chi Kin Chan, Karho Yau, Kar Hung Wong and Colin Myburgh
2013, 9(3): 505-524 doi: 10.3934/jimo.2013.9.505 +[Abstract](118) +[PDF](419.8KB)
In this paper, we consider an optimal control problem for a cleaning program involving effluent discharge of several species in a circular pool. A computational scheme combining control parametrization and finite element method is used to develop a cleaning program to meet the environmental health requirements. A numerical example is solved to illustrate the efficiency of our method.
Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs
Xinmin Yang, Jin Yang and Heung Wing Joseph Lee
2013, 9(3): 525-530 doi: 10.3934/jimo.2013.9.525 +[Abstract](55) +[PDF](276.3KB)
In this paper, we establish a strong duality theorem for Mond-Weir type multiobjective higher order nondifferentiable symmetric dual programs. Our works correct some deficiencies in recent papers [higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290(2004)423-435] and [A note on higher-order nondifferentiable symmetric duality in multiobjective programming, Appl. Math. Letters 24(2011) 1308-1311].
A conic approximation method for the 0-1 quadratic knapsack problem
Jing Zhou, Dejun Chen, Zhenbo Wang and Wenxun Xing
2013, 9(3): 531-547 doi: 10.3934/jimo.2013.9.531 +[Abstract](72) +[PDF](437.9KB)
This paper solves the 0-1 quadratic knapsack problem using a conic approximation method. We propose a nonnegative quadratic function cone program to equivalently represent the problem. Based on the technique of linear matrix inequality, we present an adaptive approximation scheme to obtain a global optimal solution or lower bound for the problem by using computable cones. Some computational examples are provided to show the effectiveness of the proposed method.
American type geometric step options
Xiaoyu Xing and Hailiang Yang
2013, 9(3): 549-560 doi: 10.3934/jimo.2013.9.549 +[Abstract](61) +[PDF](345.9KB)
The step option is a special contact whose value decreases gradually in proportional to the spending time outside a barrier of the asset price. European step options were introduced and studied by Linetsky [11] and Davydov et al. [2]. This paper considers American step options, including perpetual case and finite expiration time case. In perpetual case, we find that the optimal exercise time is the first crossing time of the optimal level. The closed price formula for perpetual step option could be derived through Feynman-Kac formula. As for the latter, we present a system of variational inequalities satisfied by the option price. Using the explicit finite difference method we could get the numerical option price.
A log-exponential regularization method for a mathematical program with general vertical complementarity constraints
Jie Zhang, Shuang Lin and Li-Wei Zhang
2013, 9(3): 561-577 doi: 10.3934/jimo.2013.9.561 +[Abstract](66) +[PDF](467.7KB)
Based on the log-exponential function, a regularization method is proposed for solving a mathematical program with general vertical complementarity constraints (MPVCC) considered by Scheel and Scholtes (Math. Oper. Res. 25: 1-22, 2000). With some known smoothing properties of the log-exponential function, a difficult MPVCC is reformulated as a smooth nonlinear programming problem, which becomes solvable by using available nonlinear optimization software. Detailed convergence analysis of this method is investigated and the results obtained generalize conclusions in Yin and Zhang (Math. Meth. Oper. Res. 64: 255-269, 2006). An example of Stackelberg game is illustrated to show the application of this method.
On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach
Vadim Azhmyakov, Alex Poznyak and Omar Gonzalez
2013, 9(3): 579-593 doi: 10.3934/jimo.2013.9.579 +[Abstract](66) +[PDF](390.4KB)
This paper is devoted to a problem of robust control design for a class of continuous-time dynamic systems with bounded uncertainties. We study a family of nonlinearly affine control systems and develop a computational extension of the conventional invariant ellipsoid techniques. The obtained method can be considered as a powerful numerical approach that makes it possible to design a concrete stabilizing control strategies for the resulting closed-loop systems. The design procedure for this feedback-type control is based on the classic Lyapunov-type stability analysis of invariant sets for the given dynamic system. We study the necessary theoretic basis and propose a computational algorithm that guarantee some minimality properties of the stability/attractivity regions for dynamic systems under consideration. The complete solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the proposed robust control design is illustrated by a numerical example.
Nonlinear conjugate gradient methods with sufficient descent properties for unconstrained optimization
Wataru Nakamura, Yasushi Narushima and Hiroshi Yabe
2013, 9(3): 595-619 doi: 10.3934/jimo.2013.9.595 +[Abstract](89) +[PDF](509.1KB)
It is very important to generate a descent search direction independent of line searches in showing the global convergence of conjugate gradient methods. The method of Hager and Zhang (2005) satisfies the sufficient descent condition. In this paper, we treat two subjects. We first consider a unified formula of parameters which establishes the sufficient descent condition and follows the modification technique of Hager and Zhang. In order to show the global convergence of the conjugate gradient method with the unified formula of parameters, we define some property (say Property A). We prove the global convergence of the method with Property A. Next, we apply the unified formula to a scaled conjugate gradient method and show its global convergence property. Finally numerical results are given.
Generalized weak sharp minima of variational inequality problems with functional constraints
Wenyan Zhang, Shu Xu, Shengji Li and Xuexiang Huang
2013, 9(3): 621-630 doi: 10.3934/jimo.2013.9.621 +[Abstract](71) +[PDF](348.5KB)
In this paper, the notion of generalized weak sharp minima is introduced for variational inequality problems with functional constraints in finite-dimensional spaces by virtue of a dual gap function. Some equivalent and necessary conditions for the solution set of the variational inequality problems to be a set of generalized weak sharp minima are obtained.
Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements
Tianliang Hou and Yanping Chen
2013, 9(3): 631-642 doi: 10.3934/jimo.2013.9.631 +[Abstract](66) +[PDF](543.9KB)
In this paper, we investigate the superconvergence property of a quadratic elliptic control problem with pointwise control constraints. The state and the co-state variables are approximated by the Raviart-Thomas mixed finite element of order $k=1$ and the control variable is discretized by piecewise linear but discontinuous functions. Approximations of the optimal solution of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order $h^{2}$.
Multi-period mean-variance portfolio selection with fixed and proportional transaction costs
Zhen Wang and Sanyang Liu
2013, 9(3): 643-656 doi: 10.3934/jimo.2013.9.643 +[Abstract](86) +[PDF](366.4KB)
Portfolio selection problem is one of the core research fields in modern financial economics. Considering the transaction costs in multi-period investments makes portfolio selection problems hard to solve. In this paper, the multi-period mean-variance portfolio selection problems with fixed and proportional transaction costs are investigated. By introducing the Lagrange multiplier and using the dynamic programming approach, the indirect utility function is defined for solving the portfolio selection problem constructed in this paper. The optimal strategies and the boundaries of the no-transaction region are obtained in the explicit form. And the efficient frontier for the original portfolio selection problems is also given. Numerical result shows that the method provided in this paper works well.
Optimality conditions for vector equilibrium problems and their applications
Adela Capătă
2013, 9(3): 659-669 doi: 10.3934/jimo.2013.9.659 +[Abstract](74) +[PDF](356.2KB)
The purpose of this paper is to establish necessary and sufficient conditions for a point to be solution of a vector equilibrium problem with cone and affine constraints. Using a separation theorem, which involves the quasi-interior of a convex set, we obtain optimality conditions for solutions of the vector equilibrium problem. Then, the main result is applied to vector optimization problems with cone and affine constraints and to duality theory.
Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces
Gang Li, Xiaoqi Yang and Yuying Zhou
2013, 9(3): 671-687 doi: 10.3934/jimo.2013.9.671 +[Abstract](74) +[PDF](390.6KB)
By using properties of dualizing parametrization functions, Lagrangian functions and the epigraph technique, some sufficient and necessary conditions of the stable strong duality and strong total duality for a class of DC optimization problems are established.
Convex hull of the orthogonal similarity set with applications in quadratic assignment problems
Yong Xia
2013, 9(3): 689-701 doi: 10.3934/jimo.2013.9.689 +[Abstract](63) +[PDF](360.2KB)
In this paper, we study thoroughly the convex hull of the orthogonal similarity set and give a new representation. When applied in quadratic assignment problems, it motivates two new lower bounds. The first is equivalent to the projected eigenvalue bound, while the second highly outperforms several well-known lower bounds in literature.
Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming
Ye Tian, Shu-Cherng Fang, Zhibin Deng and Wenxun Xing
2013, 9(3): 703-721 doi: 10.3934/jimo.2013.9.703 +[Abstract](67) +[PDF](424.1KB)
In this paper, we provide a computable representation of the cone of nonnegative quadratic forms over a general nontrivial second-order cone using linear matrix inequalities (LMI). By constructing a sequence of such computable cones over a union of second-order cones, an efficient algorithm is designed to find an approximate solution to a completely positive programming problem using semidefinite programming techniques. In order to accelerate the convergence of the approximation sequence, an adaptive scheme is adopted, and ``reformulation-linearization technique'' (RLT) constraints are added to further improve its efficiency.

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