Arnold Neumaier 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
One of the relevant research topics to which Chris Floudas contributed was quadratically constrained quadratic programming (QCQP). This paper considers one of the simplest hard cases of QCQP, the two trust region subproblem (TTRS). In this case, one needs to minimize a quadratic function constrained by the intersection of two ellipsoids. The Lagrangian dual of … Read more
This paper describes an algorithm for solving structured nonsmooth convex optimization problems using OSGA, a first-order method with the complexity $O(\eps^{-2})$ for Lipschitz continuous nonsmooth problems and $O(\eps^{-1/2})$ for smooth problems with Lipschitz continuous gradient. If the nonsmoothness of the problem is manifested in a structured way, we reformulate the problem in a form that … Read more
This paper presents an acceleration of the optimal subgradient algorithm OSGA \cite{NeuO} for solving convex optimization problems, where the objective function involves costly affine and cheap nonlinear terms. We combine OSGA with a multidimensional subspace search technique, which leads to low-dimensional problem that can be solved efficiently. Numerical results concerning some applications are reported. A … Read more
This paper shows that the OSGA algorithm — which uses first-order information to solve convex optimization problems with optimal complexity — can be used to efficiently solve arbitrary bound-constrained convex optimization problems. This is done by constructing an explicit method as well as an inexact scheme for solving the bound-constrained rational subproblem required by OSGA. … Read more
This paper shows that the optimal subgradient algorithm, OSGA, proposed in \cite{NeuO} can be used for solving structured large-scale convex constrained optimization problems. Only first-order information is required, and the optimal complexity bounds for both smooth and nonsmooth problems are attained. More specifically, we consider two classes of problems: (i) a convex objective with a … Read more
In rigorous constrained global optimization, upper bounds on the objective function help to reduce the search space. Their construction requires finding a narrow box around an approximately feasible solution, verified to contain a feasible point. Approximations are easily found by local optimization, but the verification often fails. In this paper we show that even if … Read more
A directed Cholesky factorization of a symmetric interval matrix \A consists of a permuted upper triangular matrix R such that for all symmetric A \in \A, the residual matrix A – R^T R is positive semidefinite with tiny entries. This must holds with full mathematical rigor, although the computations are done in floating-point arithmetic. Similarly, … Read more
This paper presents an algorithm for approximately minimizing a convex function in simple, not necessarily bounded convex domains, assuming only that function values and subgradients are available. No global information about the objective function is needed apart from a strong convexity parameter (which can be put to zero if only convexity is known). The worst … Read more