# Variational Inequality Methods for Engineering

The Variation Inequality (VI) problem constitutes a very general class of problems in nonlinear analysis. The VI framework embraces many different types of problems such as systems of equations, optimization problems, equilibrium programming, complementary problems, saddle-point problems, Nash equilibrium problems, and generalized Nash equilibrium problems. It specially bears strong connections with game theory. There is a well-developed theory for the analysis of solutions of VIs, as well as a wide variety of efficient algorithms with convergence properties. Therefore, it constitutes an excellent tool for analyzing the previous problems and, in particular, game problems where the classical game-theoretic tools may fall short. One of our major contribution in this context has been developing for the first time a general unified framework based on VI theory suitable to study arbitrary (generalized) Nash equilibrium problems and devise distributed asynchronous algorithms along with their convergence conditions [1,2]. We then successfully applied our machinery to a variety of problems, embracing signal processing, communications, networking, financial engineering, and smart grids [3, 4, 5, 6, 7, 8]. These applied contexts provide solid evidence of the wide applicability of the developed VI methodology.

One of my long term interests is to follow and further develop the emerging trend of merging contemporary optimization with traditional mathematics, for example to bring the ideas of dynamics into the world of static equilibrium analysis. One could say that optimal control theory does this already, but this mixes dynamics with optimization rather than with equilibrium problems. So-called Differential Variational Inequalities seems to be the
promising mathematical tool useful for this goal.

### Related Publications

1. Gesualdo Scutari, Francisco Facchinei, Jong-Shi Pang, and Daniel Palomar, “Real and Complex Monotone Communication Games,” IEEE Trans. on Information Theory, vol. 60, no. 7, pp. 4197-4231, July 2014. [PDF]
2. Gesualdo Scutari, Daniel P. Palomar, Francisco Facchinei, and Jong-Shi Pang, "Monotone Games for Cognitive Radio Systems," in Distributed Decision-Making and Control, Eds. Anders Rantzer and Rolf Johansson, Lecture Notes in Control and Information Sciences Series, Springer Verlag, 2011.
3. Gesualdo Scutari, Francisco Facchinei, Daniel P. Palomar, and Jong-Shi Pang, “Convex Optimization, Game Theory, and Variational Inequality Theory in Multiuser Communication Systems,” IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 35–49, May 2010. [Top 100 downloaded articles in IEEE among 1.25 million articles available (April-July 2010)], [Top 10 downloaded articles in IEEE SP Magazine (April-August 2010)].
4. Francisco Facchinei, Jong-Shi Pang, Gesualdo Scutari, and Lorenzo Lampariello, "VI-constrained Hemivariational Inequalities: Distributed Algorithms and Power Control in Ad-Hoc Networks," Mathematical Programming, Feb. 2013, DOI: 10.1007/s10107-013-0640-5.
5. Jong-Shi Pang, Gesualdo Scutari, Daniel P. Palomar, and Francisco Facchinei, “Design of Cognitive Radio Systems Under Temperature-Interference Constraints: A Variational Inequality Approach,” IEEE Trans. on Signal Processing, vol. 58, no. 6, pp. 3251–3271, June 2010.
6. Gesualdo Scutari, Daniel P. Palomar, Jong-Shi Pang, and Francisco Facchinei, “Flexible Design for Cognitive Wireless Systems: From Game Theory to Variational Inequality Theory,” IEEE Signal Processing Magazine, vol. 26, no. 5, pp. 107–123, Sept. 2009. [Top 100 downloaded articles in IEEE among 1.25 million articles available (Sept. 2009)], [Top 10 downloaded articles in IEEE SP Magazine (Oct. 2009)].
7. Francisco Facchinei, Jong-Shi Pang, and Gesualdo Scutari, “Non-Cooperative Games with Minmax Objectives,” Computational Optimization and Applications, Springer, to appear, (submitted, Jan. 2013, Revised July 2013) [PDF].
8. Yang Yang, Francisco Rubio, Gesualdo Scutari, and Daniel P. Palomar, “Multi-Portfolio Optimization: A Potential Game Approach,” IEEE Trans. on Signal Processing, vol. 61, no. 22, pp. 5590–5602, Nov. 2013. [Top 10 downloaded articles in IEEE TSP (Oct. 2013)].