ISSN:

2155-3289

eISSN:

2155-3297

## Numerical Algebra, Control & Optimization

2015 , Volume 5 , Issue 3

Special issue in Memory of Xunjing Li

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2015, 5(3): 233-236
doi: 10.3934/naco.2015.5.233

*+*[Abstract](2016)*+*[PDF](268.4KB)**Abstract:**

It is well known that short run cost functions of firms are convex functions when production functions are concave [14]. Average cost minimization as a classical economics problem has been studied in fundamental textbooks [14,4,7,8] and in the literature [2,3,9,12,13,1]. However, it seems that less attention so far has been paid to the study of properties of the average cost function and its minimization methods. The aim of this paper is to fulfill this gap. First, we show that average cost functions are pseudoconvex. Second, we develop an algorithm for solving the average cost minimization problem. We implement the algorithm to solve a real carpet manufacturing problem in Mongolia.

2015, 5(3): 237-249
doi: 10.3934/naco.2015.5.237

*+*[Abstract](2391)*+*[PDF](364.8KB)**Abstract:**

In this paper, a general fractional model is proposed. Based on the fractional model, a quasi-Newton trust region algorithm is presented for unconstrained optimization. The trust region subproblem is solved in the simplified way. We discussed the choices of the parameters in the fractional model and prove the global convergence of the proposed algorithm. Some primary test results shows the feasibility and validity of the fractional model.

2015, 5(3): 251-266
doi: 10.3934/naco.2015.5.251

*+*[Abstract](2101)*+*[PDF](325.5KB)**Abstract:**

In this paper, we propose a computational approach to solve a model-based optimal control problem. Our aim is to obtain the optimal solution of the nonlinear optimal control problem. Since the structures of both problems are different, only solving the model-based optimal control problem will not give the optimal solution of the nonlinear optimal control problem. In our approach, the adjusted parameters are added into the model used so as the differences between the real plant and the model can be measured. On this basis, an expanded optimal control problem is introduced, where system optimization and parameter estimation are integrated interactively. The Hamiltonian function, which adjoins the cost function, the state equation and the additional constraints, is defined. By applying the calculus of variation, a set of the necessary optimality conditions, which defines modified model-based optimal control problem, parameter estimation problem and computation of modifiers, is then derived. To obtain the optimal solution, the modified model-based optimal control problem is converted in a nonlinear programming problem through the canonical formulation, where the gradient formulation can be made. During the iterative procedure, the control sequences are generated as the admissible control law of the model used, together with the corresponding state sequences. Consequently, the optimal solution is updated repeatedly by the adjusted parameters. At the end of iteration, the converged solution approaches to the correct optimal solution of the original optimal control problem in spite of model-reality differences. For illustration, two examples are studied and the results show the efficiency of the approach proposed.

2015, 5(3): 267-274
doi: 10.3934/naco.2015.5.267

*+*[Abstract](2076)*+*[PDF](303.0KB)**Abstract:**

A dominance rule for group invertible matrices using proper splitting is proposed, and used this notion to show that a matrix is group monotone. Then some possible applications are discussed.

2015, 5(3): 275-288
doi: 10.3934/naco.2015.5.275

*+*[Abstract](2189)*+*[PDF](313.0KB)**Abstract:**

In this paper, we propose an output regulation approach, which is based on principle of model-reality differences, to obtain the optimal output measurement of a discrete-time nonlinear stochastic optimal control problem. In our approach, a model-based optimal control problem with adding the adjustable parameters is considered. We aim to regulate the optimal output trajectory of the model used as closely as possible to the output measurement of the original optimal control problem. In doing so, an expanded optimal control problem is introduced, where system optimization and parameter estimation are integrated. During the computation procedure, the differences between the real plant and the model used are measured repeatedly. In such a way, the optimal solution of the model is updated. At the end of iteration, the converged solution approaches closely to the true optimal solution of the original optimal control problem in spite of model-reality differences. It is important to notice that the resulting algorithm could give the output residual that is superior to those obtained from Kalman filtering theory. The accuracy of the output regulation is therefore highly recommended. For illustration, a continuous stirred-tank reactor problem is studied. The results obtained show the efficiency of the approach proposed.

2015, 5(3): 289-326
doi: 10.3934/naco.2015.5.289

*+*[Abstract](2639)*+*[PDF](535.7KB)**Abstract:**

This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions $A + BXB^{*}$ subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum $A + X$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function $A + BXB^*$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties $A + BXB^{*}$ from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for the equality $A + BXB^* = 0$ and the inequality $A + BXB^* \succ 0\, (\succeq 0, \prec 0, \, \preceq 0)$ to hold respectively for these specified Hermitian matrices $X$.

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