On some matrix operator and its applications

Tadeusz Ostrowski, Petroula M. Mavrikiou

Abstract


The paper focuses on a matrix operator, which maps a square real matrix to a block matrix (called the saddle point matrix), where the left-up block represents the given matrix, the right-down block is zero, and two other blocks are vectors of ones. The operator transforms any symmetric matrix into the Karush-Kuhn-Tucker matrix of standard quadratic program on the standard simplex, which is the intersection of a hyperplane with the positive orthant. There are shown some properties of this matrix operator, connections with game theory and necessary and sufficient conditions for existence of unique interior optimizer of standard quadratic program.

Keywords


Karush-Kuhn-Tucker Matrix; Constrained Optimization; Quadratic Forms; Game Theory.

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References


Bayer-Fluckinger E. Levis D. Ranicki A. Quadratic Forms and Their Applications, Series: Contempo-rary Mathematics, Vol. 272 (2001).

Bomze I. et. al. Annealed replication: a new heuristic for the maximum clique problem, Discrete Ap-plied Mathematics 121, pp. 2749 (2002).

Bomze I. and de Klerk E. Solving standard quadratic optimization problems via linear, semidefinite and copositive programming, Journal of Global Optimization 24, pp. 16385 (2002).

Boyd S. and Vandenberghe L. Convex Optimization, Cambridge University Press, UK (2004).

Cheon T. Altruistic Duality in Evolutionary Game Theory, Phys. Letters, A318, pp. 32732, (2003).

Gui-Dong Y., Yi-Zheng F. and Yi W. Quadratic forms on graphs with application to minimizing the least eigenvalue of signless Laplacian over bicyclic graphs, Electronic Journal of Linear Algebra, Vol.27. pp. 213-236 (2014).

Higham N. J. and Cheng S. H. Modifying the inertia of matrices arising in optimization, Liner Alge-bra and its Applications, Vol. 275/27, No 1-3, pp. 261-279 (1998).

Kearton, C. Quadratic Forms in knot theory, Proceedings of the Conference on Quadratic Forms and Their Applications, University College Dublin, pp. 13554 (1999).

Owen G. Game Theory, Emerald Group Publishing (2013).

Pelillo M.. Replicator Equations, Maximal Cliques, and Graph Isomorphism, Neural Computation, Vol. 11, Issue 9, pp. 193355 (1999).

Pelillo M. Siddiqi K. and Zucker S.. Continuous-based Heuristics for Graph and Tree Isomorphisms, with Application to Computer Vision, Nonconvex Optimization and its Applications, Vol 42, pp. 42245 (2000).

Trzaska Z. Applications of Quadratic Forms to Modelling and Simulations of Dynamical Systems, The International Journal of Computers, Systems and Signals, Vol. 3, No. 1, pp. 7189 (2002).

Zhang F. Edited by. The Schur Complement And Its Applications, Series: Numerical Methods and Algorithms, Vol. 4, Springer (2005).


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