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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 \(\Omega=\{x\in{\mathbb{R}^n}\mid l\leq{x}\leq{u}\}\), whereas \(f:\mathbb{R}^n\to \mathbb{R}\) 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 \((1/2)\langle{p},B^kp\rangle+\langle\nabla{f}(x^k),p\rangle\) 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.

MSC:
90C55 Methods of successive quadratic programming type
Software:
CUTEr; LANCELOT; TRON
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