Minibatch Forward-Backward-Forward Methods for Solving Stochastic Variational Inequalities

References

• Atchadé YF , Fort G , Moulines E (2017) On perturbed proximal gradient algorithms. J. Machine Learn. Res. 18(1):310–342.Google Scholar
• Bottou L , Curtis FE , Nocedal J (2018) Optimization methods for large-scale machine learning. SIAM Rev. 60(2):223–311.Google Scholar
• Boyd S , Parikh N , Chu E , Peleato B , Eckstein J (2011) Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations Trends Machine Learning 3(1):1–122.Google Scholar
• Boyd SP , Vandenberghe L (2004) Convex Optimization (Cambridge University Press, Cambridge, UK).Google Scholar
• Chen C , Chen Y , Ouyang Y , Pasiliao E (2018) Stochastic accelerated alternating direction method of multipliers with importance sampling. J. Optim. Theory Applications 179(2):676–695.Google Scholar
• Combettes P , Pesquet J (2015) Stochastic quasi-Fejér block-coordinate fixed point iterations with random sweeping. SIAM J. Optim. 25(2):1221–1248.Google Scholar
• Cottle RW , Yao JC (1992) Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Applications 75(2):281–295.Google Scholar
• Cui S , Goldsmith AJ , Bahai A (2004) Energy-efficiency of MIMO and cooperative MIMO techniques in sensor networks. IEEE J. Selected Areas Comm. 22(6):1089–1098.Google Scholar
• Dang CD , Lan G (2015) On the convergence properties of non-Euclidean extragradient methods for variational inequalities with generalized monotone operators. Comput. Optim. Applications 60(2):277–310.Google Scholar
• Duflo M (1996) Algorithmes Stochastiques (Springer, New York).Google Scholar
• Duvocelle B , Mertikopoulos P , Staudigl M , Vermeulen D (2018) Learning in time-varying games. Preprint, submitted September 10, 1918, https://arxiv.org/abs/1809.03066.Google Scholar
• Facchinei F , Pang JS (2003) Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, vols. 1 and 2 (Springer, New York).Google Scholar
• Feng D , Jiang C , Lim G , Cimini LJ Jr , Feng G , Li GY (2013) A survey of energy-efficient wireless communications. IEEE Comm. Survey and Tutorial 15(1):167–178.Google Scholar
• Hata M (1980) Empirical formula for propagation loss in land mobile radio services. IEEE Trans. Vehicular Technol. 29(3):317–325.Google Scholar
• Isheden C , Chong Z , Jorswieck E , Fettweis G (2012) Framework for link-level energy efficiency optimization with informed transmitter. IEEE Trans. Wireless Comm. 11(8):2946–2957.Google Scholar
• Iusem A , Jofré A , Oliveira RI , Thompson P (2017) Extragradient method with variance reduction for stochastic variational inequalities. SIAM J. Optim. 27(2):686–724.Google Scholar
• Jiang H , Xu H (2008) Stochastic approximation approaches to the stochastic variational inequality problem. IEEE Trans. Automatic Control 53(6):1462–1475.Google Scholar
• Jofré A , Thompson P (2019) On variance reduction for stochastic smooth convex optimization with multiplicative noise. Mathematical Programming 174:253–292.Google Scholar
• Juditsky A , Nemirovski AS , Tauvel C (2011) Solving variational inequalities with stochastic mirror-prox algorithm. Stochastic Systems 1(1):17–58.Link, Google Scholar
• Kannan A , Shanbhag U (2012) Distributed computation of equilibria in monotone nash games via iterative regularization techniques. SIAM J. Optim. 22(4):1177–1205.Google Scholar
• Kannan A , Shanbhag UV (2019) Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants. Comput. Optim. Applications 74(3):779–820.Google Scholar
• King AJ , Rockafellar RT (1993) Asymptotic theory for solutions in statistical estimation and stochastic programming. Math. Oper. Res. 18(1):148–162.Link, Google Scholar
• Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Èkonom. i Mat . Metody 12:747–756.Google Scholar
• Kushner HJ , Yin GG (1997) Stochastic Approximation Algorithms and Applications (Springer, New York).Google Scholar
• Mertikopoulos P , Belmega EV (2016) Learning to be green: Robust energy efficiency maximization in dynamic MIMO-OFDM systems. IEEE J. Selected Areas Comm. 34(4):743–757.Google Scholar
• Mertikopoulos P , Staudigl M (2018) Stochastic mirror descent dynamics and their convergence in monotone variational inequalities. J. Optim. Theory Applications 179(3):838–867.Google Scholar
• Mertikopoulos P , Zhou Z (2018) Learning in games with continuous action sets and unknown payoff functions. Mathematical Programming 173:465–507.Google Scholar
• Polyak BT (1987) Introduction to Optimization (Optimization Software, New York).Google Scholar
• Ravat U , Shanbhag U (2011) On the characterization of solution sets of smooth and nonsmooth convex stochastic nash games. SIAM J. Optim. 21(3):1168–1199.Google Scholar
• Rosasco L , Villa S , Vũ BC (2016) Stochastic forward–backward splitting for monotone inclusions. J. Optim. Theory Applications 169(2):388–406.Google Scholar
• Scutari G , Palomar DP , Facchinei F , Pang JS (2010) Convex optimization, game theory, and variational inequality theory. IEEE Signal Processing Magazine 27(3):35–49.Google Scholar
• Shapiro A , Dentcheva D , Ruszczyński AX (2009) Lectures on Stochastic Programming: Modeling and Theory (SIAM, Philadelphia). Google Scholar
• Shen K , Yu W (2018) Fractional programming for communication systems: I. Power control and beamforming. IEEE Trans. Signal Processing 66(10):2616–2630.Google Scholar
• Solodov M , Svaiter B (1999) A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3):765–776.Google Scholar
• Stroock DW (2011) Probability Theory: An Analytic View , 2nd ed. (Cambridge University Press, Cambridge, UK).Google Scholar
• Telatar IE (1999) Capacity of multi-antenna Gaussian channels. Eur. Trans. Telecomm. Related Tech . 10(6):585–596.Google Scholar
• Tseng P (2000) A modified forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2):431–446.Google Scholar
• Von Stengel B (2002) Computing equilibria for two-person games. Handbook of Game Theory with Economic Applications, vol. 3 (Elsevier, Amsterdam), 1723–1759.Google Scholar
• Yousefian F , Nedić A , Shanbhag UV (2017) On smoothing, regularization, and averaging in stochastic approximation methods for stochastic variational inequality problems. Math. Programming 165(1):391–431.Google Scholar