Approximate Optimal Value Research Articles

Intensity modulated radiotherapy treatment planning by use of a barrier-penalty multiplier method

The use of nonlinear functions describing biological effects has recently become a major goal in connection with intensity modulated radiotherapy planning models for cancer treatment. In this article, we present a new biological model for this purpose and discuss the solution of the related large-scale nonlinear optimization problems. The model includes equivalent uniform dose and partial volume constraints and employs tumor control probability as the objective. The resulting optimization problems are convex; there are nonconvex constrained optimization problems with several thousands of variables for which gradients of the involved functions are available, but the computation of Hessians is too costly. It is suggested to solve these problems using the barrier-penalty multiplier method by Polyak ([Polyak, R., 1992, Modified barrier functions (theory and methods). Mathematical Programming, 54, 177–222.], [Polyak, R., 2002, Nonlinear rescaling vs. smoothing technique in convex optimization. Mathematical Programming, 92, 197–235.]) and Ben-Tal et al. and Ben-Tal and Zibulevsky ([Ben-Tal, A., Yuzefokich, I. and Zibulevsky, M., 1992, Penalty/barrier multiplier methods for minimax and constrained smooth convex problems. Technical Report 9/92, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Haifa, Israel.], [Ben-Tal, A. and Zibulevsky, M., 1997, Penalty/barrier multiplier methods for convex programming problems. SIAM Journal of Optimization, 7, 347–366.]), where this algorithm is modified according to ideas which are motivated by the related Lagrangian barrier algorithm of Conn et al. ([Conn, A.R., Gould, N.I.M. and Taint, P., 1992, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Technical Report 92/07, Department of Maths, FUNDP, Namur, Belgium.], [Conn, A.R., Gould, N.I.M. and Taint, P.L., 1992, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Mathematics of Computation, 66, 261–288.]). In particular, the subproblems in the algorithm are solved by a conjugate gradient method, as the spectrum of the Hessian of the Lagrangian at a solution of such problem indicates fast (local) convergence of the objective function values to a good approximate (locally) optimal value. Some characteristic numbers showing the average numerical performance of the algorithm are tabulated for various types of tumors and for a set of 127 clinical cases in total. Its capabilities and typical behavior also are illustrated explicitly by a computed therapy plan for a difficult clinical case of a larynx tumor.

Constructing Robust Feedback Laws by Set Oriented Numerical Methods

AbstractIn [8, 6] a numerical method for the construction of optimally stabilizing feedback laws was proposed. The method is based on a set oriented discretization of phase space in combination with graph theoretic algorithms for the computation of shortest paths in directed weighted graphs. The resulting approximate optimal value function is piecewise constant, yielding an approximate optimal feedback which might not be robust with respect to perturbations of the system. In this contribution we extend the approach to the case of perturbed control systems. Based on the concept of a multivalued game we show how to derive a directed weighted hypergraph from the original system and generalize the corresponding shortest path algorithm. The resulting optimal value function yields a robustly stabilizing approximate optimal feedback law. This note is an abbreviated version of [5]. For the proofs of the statements here we refer to the full paper. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Study on cottonseed oil as a partial substitute for diesel oil in fuel for single-cylinder diesel engine

The experiments were undertaken to obtain the knowledge necessary for raising the thermal efficiency of mixed oil composed of cottonseed oil and conventional diesel oil and for improving the performance of engine fuelled by the mixture. The experimental results obtained showed that a mixing ratio of 30% cottonseed oil and 70% diesel oil was practically optimal in ensuring relatively high thermal efficiency of engine, as well as homogeneity and stability of the oil mixture. A quadratic regressive orthogonal design test method was adopted in the experiment designed to examine the relationship between specific fuel consumption and four adjustable working parameters (intake-valve-closing angle ( α), exhaust-valve-opening angle ( β), fuel-delivery angle ( θ) and injection pressure ( P, in 10 4 Pa)) when the above-mentioned oil mixture was used. The mathematical equations characterizing the relationship were formulated. The equation of specific fuel consumption derived from the regressive test under each operating condition was set as the objective function and the ranges for the four adjustable working parameters were the given constraint condition. Models of non-linear programming were then constructed. Computer-aided optimization of the working parameters for 30:70 cottonseed oil/diesel oil mixed fuel was achieved. It was concluded that the predominant factor affecting the specific fuel consumption was fuel-delivery angle θ, the approximate optimal value of which, in this specific case, was 3–5° in advance of that for engine fuelled by pure diesel oil. The experimental results also provided useful reference material for selection of the most preferable combination of working parameters.

Study on rapeseed oil as alternative fuel for a single-cylinder diesel engine

This study was undertaken to provide knowledge necessary for raising the thermal efficiency of mixed oil composed of rapeseed oil and conventional diesel oil and for improving the performance of an engine fuelled by the mixture. The experimental results obtained showed that a mixing ratio of 30% rapeseed oil and 70% diesel oil was practically optimal in ensuring relatively high thermal efficiency of engine as well as homogeneity and stability of the oil mixture. Method of quadratic regressive orthogonal design test method was adopted in experiment designed to examine the dependence of specific fuel consumption on four adjustable working parameters when the above –mentioned oil mixture was used. These parameters were: intake-valve-closing angle ( α), exhaust-valve-opening angle ( β), fuel-delivering angle ( θ) and injection pressure ( P, in 10 4 Pa). Relationship between these parameters and specific fuel consumption was analyzed under two typical operating conditions and mathematical equations characterizing the relationship were formulated. The equation of specific fuel consumption derived from the regressive test under each operating condition was set as the objective function and the ranges for the four adjustable working parameters were the given constraint condition. Models of non-linear programming were then constructed. Computer aided optimization of the working parameters for 30:70 rapeseed oil/diesel oil mixed fuel was achieved. It was concluded that the predominant factor affecting the specific fuel consumption was fuel-delivering angle θ, the approximate optimal value of which, in this specific case, was 2–3 degrees in advance of that for engine fuelled by pure diesel oil. The experimental results also provided useful reference material for selection of the most preferable combination of working parameters.

Approximate Optimal Solutions and Nonlinear Lagrangian Functions

There is an increasing interest in the study of optimality conditions of approximate solutions for nonlinear optimization problems. In this paper, relationships between approximate optimal values and approximate roots of a nonlinear function are explored via a nonlinear Lagrangian function. Almost approximate optimal solutions are investigated by means of nonlinear Lagrangian functions.

ジャンプフラッシング放電加工の等価モデル （第１報） 電気回路によるモデル表現

In the diesinking EDM, the jump flushing is generally known to be useful to clean up the working gap. However, it is very difficult to adjust the jump parameters efficiently such as an amplitude and an execution timing of the jump motion with progress of machining. Because, in this process, optimal values of such parameters change widely with progress of machining and to adjust them an extensive expert knowledge about the jump control is required. Therefore, if a numerical equivalent model of the jump flushing EDM can be framed, it will be very useful for adaptive adjustment of the jump parameters.This report describes a newly proposed equivalent model of the jump flushing EDM represented by an electrical way. Increasing process of the machining products in the working gap is converted to the charging process of a condenser in a DC electric circuit. Efficient jump timing is able to be computed from the rising time of the terminal voltage of the condenser to a threshold value which corresponds to critical density of the machining products for the safety sparking.Results of computer simulations based on the proposed model show that the machining characteristics of the jump flushing EDM could be reproduced sufficiently except for the very early stage of machining. Besides, it was found that approximate optimal values of the jump parameters could be computed easily.

The Effects of the Asselin Time Filter on Numerical Solutions to the Linearized Shallow-Water Wave Equations

In the present investigation, a one-dimensional linearized analysis is used to determine the effect of Asselin's (1972) time filter on both the computational stability and phase error of numerical solutions for the shallow water wave equations, in cases with diffusion but without rotation. An attempt has been made to establish the approximate optimal values of the filtering parameter nu for each of the 'lagged', Dufort-Frankel, and Crank-Nicholson diffusion schemes, suppressing the computational wave mode without materially altering the physical wave mode. It is determined that in the presence of diffusion, the optimum filter length depends on whether waves are undergoing significant propagation. When moderate propagation is present, with or without diffusion, the Asselin filter has little effect on the spatial phase lag of the physical mode for the leapfrog advection scheme of the three diffusion schemes considered.

Computation of (S, s) Ordering Policy Parameters

In this paper a periodic review stock system using the so-called (S, s) ordering policy is considered. New and more accurate approximations are found for calculating the optimal values of the reorder point s and the minimum order quantity Δ = S − s. When demands are normally distributed it is shown that the six cost and distribution parameters associated with these approximations can be combined into two effective parameters. This facilitates the derivation of two graphs which can be used with some relatively simple computational procedures to determine approximate optimal values for the reorder point s and minimum order quantity Δ.