Arnold Neumaier In this paper, we introduce a new effective matrix adaptation evolution strategy (MADFO) for noisy derivative-free optimization problems. Like every MAES solver, MADFO consists of three phases: mutation, selection and recombination. MADFO improves the mutation phase by generating good step sizes, neither too small nor too large, that increase the probability of selecting mutation points … Read more
This paper defines an efficient subspace method, called SSDFO, for unconstrained derivative-free optimization problems where the gradients of the objective function are Lipschitz continuous but only exact function values are available. SSDFO employs line searches along directions constructed on the basis of quadratic models. These approximate the objective function in a subspace spanned by some … Read more
This paper introduces a new nonlinear conjugate gradient method using any efficient line search method. Unless function values diverge to $-\infty$, global convergence to a stationary point is proved for continuously differentiable objective functions with Lipschitz continuous gradient, and global linear convergence if this stationary point is a strong local minimizer. Complexity bounds are given … Read more
This paper introduces a new line search along an arbitrary smooth search path that starts at the current iterate tangentially to a descent direction. Like the Goldstein line search and unlike the Wolfe line search, the new line search uses, beyond the gradient at the current iterate, only function values. Using this line search with … Read more
This paper discusses a composite algorithm for bound constrained noisy derivative-free optimization problems with integer variables. This algorithm is an integer variant of the matrix adaptation evolution strategy. An integer derivative-free line search strategy along affine scaling matrix directions is used to generate candidate points. Each affine scaling matrix direction is a product of the … Read more
Recently, Neumaier and Azmi gave a comprehensive convergence theory for a generic algorithm for bound constrained optimization problems with a continuously differentiable objective function. The algorithm combines an active set strategy with a gradient-free line search CLS along a piecewise linear search path defined by directions chosen to reduce zigzagging. This paper describes LMBOPT, an … Read more
The first part of this paper discusses convergence properties of a new line search method for the optimization of continuously differentiable functions with Lipschitz continuous gradient. The line search uses (apart from the gradient at the current best point) function values only. After deriving properties of the new, in general curved, line search, global convergence … Read more
For the unconstrained optimization of black box functions, this paper introduces a new randomized algorithm called VRBBO. In practice, VRBBO matches the quality of other state-of-the-art algorithms for finding, in small and large dimensions, a local minimizer with reasonable accuracy. Although our theory guarantees only local minimizers our heuristic techniques turn VRBBO into an efficient … Read more
This paper presents rigorous filtering methods for constraint satisfaction problems based on the interval union arithmetic. Interval unions are finite sets of closed and disjoint intervals that generalize the interval arithmetic. They allow a natural representation of the solution set of interval powers, trigonometric functions and the division by intervals containing zero. We show that … Read more
Let $r_{n}$ be the radius of the smallest circle into which one can pack $n$ non-overlapping unit squares that are free to rotate. For $n = 1, 2$, $r_{1} = \frac{\sqrt{2}}{2}$ and $r_{2} = \frac{\sqrt{5}}{2}$ are proved to be optimal. For $n \geq 3$ only guesses of $r_{n}$ are known. This paper introduces a computer-assisted … Read more