In this work, we propose a new scheme for scalarization in set optimization studied with the Kuroiwa set appoach. First, we define an abstract scalarizing function possessing properties such as global Lipschizity, sublinearity, cone monotonicity, cone representation property, cone interior representation property and uniform positivity. Next, we use this function to define the so called … Read more
For a finite set $X \subset \Z^d$ that can be represented as $X = Q \cap \Z^d$ for some polyhedron $Q$, we call $Q$ a relaxation of $X$ and define the relaxation complexity $\rc(X)$ of $X$ as the least number of facets among all possible relaxations $Q$ of $X$. The rational relaxation complexity $\rc_\Q(X)$ restricts … Read more
The sharp local minimality of feasible points of nonlinear optimization problems is known to possess a characterization by a strengthened version of the Karush-Kuhn-Tucker conditions, as long as the Mangasarian-Fromovitz constraint qualification holds. This strengthened condition is not easy to check algorithmically since it involves the topological interior of some set. In this paper we … Read more
Bilevel optimization is a very active field of applied mathematics. The main reason is that bilevel optimization problems can serve as a powerful tool for modeling hierarchical decision making processes. This ability, however, also makes the resulting problems challenging to solve—both in theory and practice. Fortunately, there have been significant algorithmic advances in the field … Read more
Discrete Optimal Transport problems give rise to very large linear programs (LP) with a particular structure of the constraint matrix. In this paper we present an interior point method (IPM) specialized for the LP originating from the Kantorovich Optimal Transport problem. Knowing that optimal solutions of such problems display a high degree of sparsity, we … Read more
An essential component in modern solvers for mixed-integer (linear) programs (MIPs) is the separation of additional inequalities (cutting planes) to tighten the linear programming relaxation. Various algorithmic decisions are necessary when integrating cutting plane methods into a branch-and-bound (B&B) solver as there is always the trade-off between the efficiency of the cuts and their costs, … Read more
\(\) In 1994, Kranakis et al. published a conjecture about the minimum link-length of every rectilinear covering path for the \(k\)-dimensional grid \(P(n,k) := \{0,1, \dots, n-1\} \times \{0,1, \dots, n-1\} \times \cdots \times \{0,1, \dots, n-1\}\). In this paper we consider the general, NP-complete, Line-Cover problem, where the edges are not required to be … Read more
In air transportation, container crossdocking refers to a loaded container that is transferred at an airport from an incoming flight to an outgoing flight without handling the freight on the container. It reduces handling time and handling cost relative to unloading the container and sorting the freight, and is an economical alternative if a sufficient … Read more
An algorithm for finding a first-order and second-order critical point of composite nonsmooth problems is proposed in this paper. For smooth problems, algorithms for searching such a point usually utilize the so called negative-curvature directions. In this paper, the method recently proposed for nonlinear semidefinite problems by the current author is extended for solving general … Read more
Nonconvex minimization algorithms often benefit from the use of second-order information as represented by the Hessian matrix. When the Hessian at a critical point possesses negative eigenvalues, the corresponding eigenvectors can be used to search for further improvement in the objective function value. Computing such eigenpairs can be computationally challenging, particularly if the Hessian matrix … Read more