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The Vertex Separator Problem (VSP) for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets of roughly equal size. In a recent paper (Optimality Conditions For Maximizing a Function Over a Polyhedron, Mathematical Programming, 2013, doi: 10.1007/s10107-013-0644-1), the authors announced a new… (More)

This paper considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP is developed for solving this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically in… (More)

- William W. Hager, James T. Hungerford
- Math. Program.
- 2014

We present new first and second-order optimality conditions for maximizing a function over a polyhedron. These conditions are expressed in terms of the first and second-order directional derivatives along the edges of the polyhedron, and an edge description of the polyhedron. If the objective function is quadratic and edgeconvex, and the constraint… (More)

- William W. Hager, James T. Hungerford, Ilya Safro
- ArXiv
- 2014

The Vertex Separator Problem for a graph is to find the smallest collection of vertices whose removal breaks the graph into two disconnected subsets that satisfy specified size constraints. The Vertex Separator Problem was formulated in the paper 10.1016/j.ejor.2014.05.042 as a continuous (non-concave/non-convex) bilinear quadratic program. In this paper,… (More)

- William W. Hager, James T. Hungerford
- European Journal of Operational Research
- 2015

Four NP-hard optimization problems on graphs are studied: The vertex separator problem, the edge separator problem, the maximum clique problem, and the maximum independent set problem. We show that the vertex separator problem is equivalent to a continuous bilinear quadratic program. This continuous formulation is compared to known continuous quadratic… (More)

- Timothy A. Davis, William W. Hager, James T. Hungerford
- ACM Trans. Math. Softw.
- 2016

This article considers the problem of minimizing a convex, separable quadratic function subject to a knapsack constraint and a box constraint. An algorithm called NAPHEAP has been developed to solve this problem. The algorithm solves the Karush-Kuhn-Tucker system using a starting guess to the optimal Lagrange multiplier and updating the guess monotonically… (More)

We consider the vertex separator problem on a graph: find a set of vertices of minimum cost whose removal disconnects the graph into two roughly equal sized components. In this talk, we present a multilevel algorithm for the VSP, whose refinement phase is based on the solution of a bilinear program which is shown to approximate the VSP at each level in the… (More)

- William W. Hager, James T. Hungerford, Ilya Safro
- LION
- 2014

The Vertex Separator Problem (VSP) on a graph is the problem of finding the smallest collection of vertices whose removal separates the graph into two disjoint subsets of roughly equal size. Recently, Hager and Hungerford [1] developed a continuous bilinear programming formulation of the VSP. In this paper, we reinforce the bilinear programming approach… (More)

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