Browsing by Subject "simultaneous equations"

Now showing 1 - 4 of 4

Access status: Open Access ,
A loss function for box-constrained inverses problems
(2008) Yoneda, Kiyoshi
A loss function is proposed for solving box-constrained inverse problems. Given causality mechanisms between inputs and outputs as smooth functions, an inverse problem demands to adjust the input levels to make the output levels as close as possible to the target values; box-constrained refers to the requirement that all outcome levels remain within their respective permissible intervals. A feasible solution is assumed known, which is often the status quo. We propose a loss function which avoids activation of the constraints. A practical advantage of this approach over the usual weighted least squares is that permissible outcome intervals are required in place of target importance weights, facilitating data acquisition. The proposed loss function is smooth and strictly convex with closed-form gradient and Hessian, permitting Newton family algorithms. The author has not been able to locate in the literature the Gibbs distribution corresponding to the loss function. The loss function is closely related to the <i>generalized matching law</i> in psychology.
Access status: Open Access ,
A utility function to solve approximate linear equations for decision making
(2013) Yoneda, Kiyoshi; Celaschi, Walter
Suppose there are a number of decision variables linearly related to a set of outcome variables. There are at least as many outcome variables as the number of decision variables since all decisions are outcomes by themselves. The quality of outcome is evaluated by a utility function. Given desired values for all outcome variables, decision making reduces to »solving« the system of linear equations with respect to the decision variables; the solution being defined as decision variable values such that maximize the utility function. This paper proposes a family of additively separable utility functions which can be defined by setting four intuitive parameters for each outcome variable: the desired value of the outcome, the lower and the upper limits of its admissible interval, and its importance weight. The utility function takes a nonnegative value within the admissible domain and negative outside; permits gradient methods for maximization, is designed to have a small dynamic range for numerical computation. Small examples are presented to illustrate the proposed method.
Access status: Open Access ,
Maximization of an asymmetric utility function by the least squares
(2014) Yoneda, Kiyoshi; Moretti, Antonio Carlos
This note points out that a utility maximization procedure proposed in an earlier paper may be reduced to the least squares. The utility function is asymmetric in the sense that for each cue its ideal value and the permissible range are assigned in such a way that the ideal is not necessarily at the center of the range, like »a beer of 350 ml would be ideal, but acceptable if within [100, 500]«. A practical consequence of the observation is that very little programming will be needed to deploy the utility maximization since software for the least squares is widely available.
Access status: Open Access ,
Stretching the least squares to embed loss function tables
(2015) Yoneda, Kiyoshi; Moretti, Antonio Carlos; Poker, Johan Hendrik