Abstract
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 edge-convex, and the constraint polyhedron includes a box constraint, then local optimality can be checked in polynomial time. The theory is applied to continuous formulations of the vertex and edge separator problems. It is seen that the necessary and sufficient optimality conditions for these problems are related to the existence of edges at specific locations in the graph.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1999)
Borwein, J.: Necessary and sufficient conditions for quadratic minimality. Numer. Funct. Anal. Optim. 5, 127–140 (1982)
Contesse, L.: Une caractérisation complète des minima locaux en programmation quadratique. Numer. Math. 34, 315–332 (1980)
Dantzig, G.: Maximization of a linear function of variables subject to linear inequalities. In: Koopman, T.C. (eds), Activity Analysis of Production and Allocation, Cowles Commission Monograph, vol. 13, Chap. XXI. Wiley, New York (1951)
Gritzmann, P., Sturmfels, B.: Minkowski addition of polytopes: computational complexity and applications to Gröbner bases. SIAM J. Disc. Math. 6, 246–269 (1993)
Griva, I., Nash, S.G., Sofer, A.: Linear and Nonlinear Optimization. SIAM, Philadelphia (2009)
Hager, W.W., Hungerford, J.T.: A Continuous Quadratic Programming Formulation of the Vertex Separator Problem. Technical Report. Department of Mathematics, University of Florida. http://www.math.ufl.edu/~hager/papers/GP/vertex.pdf (2012)
Hager, W.W., Krylyuk, Y.: Graph partitioning and continuous quadratic programming. SIAM J. Disc. Math. 12, 500–523 (1999)
Hwang, F., Rothblum, U.: Directional-quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions. Math. Oper. Res. 21, 540–554 (1996)
Klee, V.: Some characterizations of convex polyhedra. Acta Math. 102, 79–107 (1959)
Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming. Springer, Berlin (2008)
Murty, K.G., Kabadi, S.N.: Some NP-complete problems in quadratic and linear programming. Math. Program. 39, 117–129 (1987)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)
Onn, S., Rothblum, U.: Convex combinatorial optimization. Discret. Comput. Geom. 32, 549–566 (2004)
Onn, S., Rothblum, U., Tangir, Y.: Edge-directions of standard polyhedra with applications to network flows. J. Glob. Optim. 33, 109–122 (2005)
Pardalos, P.M., Schnitger, G.: Checking local optimality in constrained quadratic programming can be NP-hard. Oper. Res. Lett. 7, 33–35 (1988)
Rockafellar, R.T.: Network Flows and Monotropic Optimization. Wiley, New York (1984)
Tardella, F.: On the equivalence between some discrete and continuous optimization problems. Ann. Oper. Res. 25, 291–300 (1990)
Tuy, H.: Concave programming under linear constraints. Soviet Math. 5, 1439–1440 (1964)
Author information
Authors and Affiliations
Corresponding author
Additional information
This material is based upon work supported by the National Science Foundation under Grants 0619080 and 0620286, and the Office of Naval Research under Grant N00014-11-1-0068.
Rights and permissions
About this article
Cite this article
Hager, W.W., Hungerford, J.T. Optimality conditions for maximizing a function over a polyhedron. Math. Program. 145, 179–198 (2014). https://doi.org/10.1007/s10107-013-0644-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-013-0644-1
Keywords
- Optimality conditions
- Critical cone
- Quadratic programming
- Edge-convexity
- Vertex separator
- Graph partitioning