The optimal design of electrical machines can be mathematically modeled as a mixed-integer nonlinear optimization problem. We present six variants of such a problem, and we show, through extensive computational experiments, that, even though they are mathematically equivalent, the differences in the formulations may have an impact on the numerical performances of a local optimization … Read more
Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of … Read more
Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone of all completely positive symmetric nxn matrices, and its … Read more
We propose two new lower bounds on graph bandwidth and cyclic bandwidth based on semidefinite programming (SDP) relaxations of the quadratic assignment problem. We compare the new bounds with two other SDP bounds in [A. Blum, G. Konjevod, R. Ravi, and S. Vempala, Semi-definite relaxations for minimum bandwidth and other vertex-ordering problems, Theoretical Computer Science, … Read more
We describe three algorithms for solving differentiable convex optimization problems constrained to simple sets in $ \R^n $, i.e., sets on which it is easy to project an arbitrary point. The first two algorithms are optimal in the sense that they achieve an absolute precision of $ \varepsilon $ in relation to the optimal value … Read more
We present a semide nite programming (SDP) approach for the problem of ordering vertices of a layered graph such that the edges of the graph are drawn as vertical as possible. This Multi-Level Vertical Ordering (MLVO) problem is a quadratic ordering problem and conceptually related to the well-studied problem of Multi-Level Crossing Minimization (MLCM). In contrast … Read more
In traditional multi-level graph drawing – known as Sugiyama’s framework – the number of crossings is considered one of the most important goals. Herein, we propose the alternative concept of optimizing the verticality of the drawn edges. We formally specify the problem, discuss its relative merits, and show that drawings that are good w.r.t. verticality … Read more
In the paper, we propose a bilevel direct search method for solving a type of leader-follower problems with each decision maker’s objective being a “black-box” function. First, we give a description for a leader-follower optimization problem. Then, we investigate a bilevel direct search method including two algorithms for combinatorially solving the upper and lower level … Read more
We propose a modification to the (generalized) bundle scheme for minimization of a convex nondifferentiable sum-function in the case where some of the components are “easy”, that is, they are Lagrangian functions of explicitly known convex programs with “few” variables and constraints. This happens in many practical cases, particularly within applications to combinatorial optimization. In … Read more
A common structure in convex mixed-integer nonlinear programs is additively separable nonlinear functions consisting of a sum of univariate functions. In the presence of such structures, we propose three improvements to the classical Outer Approximation algorithms that exploit separability. The first improvement is a simple extended formulation. The second a refined outer approximation. Finally, the … Read more