## 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

### Keywords:

box constrained optimization

### Software:

LANCELOT; CUTEr; TRON
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