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A limited-memory multipoint symmetric secant method for bound constrained optimization. (English) Zbl 1025.90038
The paper concerns a new iterative algorithm for the minimization of $f\left(x\right)$ subject to the box-constraint $x\in {\Omega }$. Here ${\Omega }=\left\{x\in {ℝ}^{n}\mid l\le x\le u\right\}$, whereas $f:{ℝ}^{n}\to ℝ$ is a continuously differentiable function. Given an iteration ${x}^{k}\in {\Omega }$, the new iteration ${x}^{k+1}\in {\Omega }$ is computed through the minimization of $\left(1/2\right)〈p,{B}^{k}p〉+〈\nabla f\left({x}^{k}\right),p〉$ subject to $p\in \overline{{F}_{I}}$. Here ${F}_{I}$ is the ${\Omega }$-face which contains ${x}^{k}$, whereas ${B}^{k}$ is a symmetric approximation of the Hessian of $f$. The matrices ${B}^{k}$ are generated by a multipoint symmetric secant method. The involved limited-memory formulae differ from the conventional ones because they are based on different quasi-Newton methods. Implementation details, numerical results, and final conclusions are presented.
##### MSC:
 90C55 Methods of successive quadratic programming type
##### Keywords:
box constrained optimization
TRON; LANCELOT