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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 $f(x)$ subject to the box-constraint $x\in\Omega$. Here $\Omega=\{x\in{\bbfR^n}\mid l\le{x}\le{u}\}$, whereas $f:\bbfR^n\to \bbfR$ 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 $(1/2)\langle{p},B^kp\rangle+\langle\nabla{f}(x^k),p\rangle$ 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.

90C55Methods of successive quadratic programming type
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