In this paper, the problem of extremizing a functionf(x) subject to the constraint ϕ(x)=0 is considered. Here,f is a scalar,x ann-vector, and ϕ aq-vector. A modified quasilinearization algorithm is developed; its main property is a descent property in the performance indexR, the cumulative error in the constraint and the optimum condition.
Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of a scaling factor (or stepsize) α in the system of variations. The stepsize α is determined by a one-dimensional search so as to ensure the decrease in the performance indexR; this can be achieved through a bisection process starting from α=1. Convergence is achieved whenR becomes smaller than some preselected value.
In order to start the algorithm, some nominal values for the variablex and the multiplier λ must be chosen. In a real problem, the selection ofx can be made on the basis of physical considerations. Concerning λ, no useful guideline has been available thus far. In this paper, a method for selecting λ optimally is presented: the performance indexR is minimized with respect to λ. SinceR is a quadratic function of λ, the optimal initial multiplier is governed by a linear algebraic equation.
Two numerical examples are presented, and it is shown that, if the initial multiplier is chosen optimally, modified quasilinearization converges to the solution. On the other hand, if the initial multiplier is chosen arbitrarily, modified quasilinearization may or may not converge to the solution. From the examples, it is concluded that the beneficial effect associated with the optimal initial choice of the multiplier lies primarily in increasing the likelihood of convergence rather than accelerating convergence. However, this optimal choice does not guarantee convergence, since convergence depends on the functionf(x), the constraint ϕ(x), and the initialx chosen in order to start the algorithm.
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This research, supported by the National Science Foundation, Grant No. GP-18522, is based on Ref. 1.
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Miele, A., Levy, A.V. Modified quasilinearization and optimal initial choice of the multipliers part 1—Mathematical programming problems. J Optim Theory Appl 6, 364–380 (1970). https://doi.org/10.1007/BF00932583
- Beneficial Effect
- Optimum Condition
- Programming Problem
- Algebraic Equation
- Mathematical Programming