EE 609: Convex Optimization in SP/COM

  1. Instructors: Ketan Rajawat
  2. Prerequisites: Linear Algebra, Probability
  3. Objective: Convex optimization has recently been applied to a wide variety of problems in EE, especially in signal processing, communications, and networks. The aim of this course is to train the students in application and analysis of convex optimization problems in signal processing and wireless communications. At the end of this course, the students are expected to:
    1. Know about the applications of convex optimization in signal processing, wireless communications, and networking research.
    2. Be able to recognize convex optimization problems arising in these areas.
    3. Be able to recognize ‘hidden’ convexity in many seemingly non-convex problems; formulate them as convex problems. 
    4. Be able to develop low-complexity, approximate solutions for difficult non-convex problems.
  4. References:
    1. Stephen Boyd and Lieven Vandenberghe, Convex Optimization, Cambridge University Press. [Online]. http://www.stanford.edu/~boyd/cvxbook/
    2. Convex Optimization in Signal Processing and Communications, D. P. Palomar, Y. C. Eldar. Cambridge Press, 2010.
    3. IEEE Signal Processing Magazine- Special Issue on Advances in Convex Optimization, Vol. 27, No. 3, May 2010.
    4. Dimitri P. Bertsekas, Convex Analysis and Optimization, Athena-Scientific, 2003.
  5. Format (tentative)
    1. Major quiz (10) scheduled on Jan. 28 (Tuesday) at 12noon; venue TBA
    2. Mid-sem exam (20) scheduled on 17 - 22 Feb  
    3. End-sem exam (35) scheduled on 20 - 29 Apr
    4. 7 Assignments (10):  valid attempts (correct or incorrect) will receive full credit
    5. 1 Term paper (25)
    6. Assignment discussions on 18 Jan, 25 Jan, 1 Feb, 12 Feb, 28 Mar (tentative)
  6. Time and place: Tuesday, Wednesday, 12 noon, L-11
  7. This course will cover (approximately) 
    1. Background on linear algebra
    2. Convex sets, functions, and problems
    3. Examples of convex problems: LP, QCQP, SOCP
    4. Duality, KKT conditions
    5. Geometric programming and applications
    6. Linear and quadratic classification
    7. Network optimization
    8. Sparse regression, Lasso, ridge regression and applications in image processing
    9. Robust least squares and applications in signal processing
    10. Support vector machines and applications in machine learning
    11. Semidefinite programming and applications in experiment design
    12. Semidefinite relaxation and applications in MIMO detection, integer programming
    13. Low rank matrix completion and applications in recommendor systems
    14. Multidimensional scaling and applications in sensor localization
    15. Numerical linear algebra, basics of interior point methods
  8. Tentative list of topics for Term Paper (only sample papers are provided, others can also be chosen, specify: theory/mixed/application mode) 
    1. Online learning and online convex optimization 
      1. Shai Shalev-Shwartz, Foundations and Trends in ML, 4(2), 2011 
      2. Introduction to Online optimization 
      3. Elad Hazan, Foundations and Trends in Optimization, 2(3-4), 2015 
    2. Proximal Algorithms 
      1. Neal Parikh and Stephen Boyd (2014), "Proximal Algorithms", Foundations and Trends in Optimization: Vol. 1: No. 3, pp 127-239 
    3. Multi-period investment 
      1. Stephen Boyd, Mark T. Mueller, Brendan O’Donoghue and Yang Wang (2014), "Performance Bounds and Suboptimal Policies for Multi–Period Investment", Foundations and Trends in Optimization: Vol. 1: No. 1 
      2. Stephen Boyd, Enzo Busseti, Steve Diamond, Ronald N. Kahn, Kwangmoo Koh, Peter Nystrup and Jan Speth (2017), "Multi-Period Trading via Convex Optimization", Foundations and Trends in Optimization: Vol. 3: No. 1 
    4. ADMM
      1. Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato and Jonathan Eckstein (2011), "Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers", Foundations and Trends in Machine Learning: Vol. 3: No. 1.
    5. Submodularity and convexity
      1. Francis Bach (2013), "Learning with Submodular Functions: A Convex Optimization Perspective", Foundations and Trends in Machine Learning: Vol. 6: No. 2-3.
    6. Learning over Networks
      1. Ali H. Sayed (2014), "Adaptation, Learning, and Optimization over Networks", Foundations and Trends® in Machine Learning: Vol. 7: No. 4-5, pp 311-801.
    7. S-lemma
      1. Polik, I., Terlaky T., A Survey on the S-Lemma , SIAM Review Vol. 49, No. 3, pp. 371-418, 2004.
      2. Derinkuyu, K., Pinar, M., On the S-Procedure and Some Variants , 2004.
    8. Approximate dynamic programming in transportation/logistics
      1. W. B. Powell, H. Simao, B. Bouzaiene-Ayari, “Approximate Dynamic Programming in Transportation and Logistics: A Unified Framework,” European J. on Transportation and Logistics, Vol. 1, No. 3, pp. 237-284 (2012). DOI 10.1007/s13676-012-0015-8.
    9. Sequential stochastic optimization, dynamic programming
      1. http://castlelab.princeton.edu/html/Papers/Powell-UnifiedFrameworkStochasticOptimization_July222017.pdf
      2. Clearing the Jungle of Stochastic Optimization, Informs Tutorials in Operations Research: Bridging Data and Decisions, pp. 109-137, November, 2014
    10. Robust optimization in logistics (sample papers only, others can be used):
      1. Bertsimas D., Thiele A. (2004) A Robust Optimization Approach to Supply Chain Management. In: Bienstock D., Nemhauser G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg
      2. Bertsimas, Dimitris, and Aurelie Thiele. "A robust optimization approach to inventory theory." Operations research 54.1 (2006): 150-168.
      3. Ben-Tal, Aharon, et al. "Retailer-supplier flexible commitments contracts: A robust optimization approach." Manufacturing & Service Operations Management 7.3 (2005): 248-271.
      4. Bertsimas, Dimitris, and Melvyn Sim. "Tractable approximations to robust conic optimization problems." Mathematical programming 107.1-2 (2006): 5-36.
    11. Subgradient methods
      1. S. Boyd and A. Mutapcic, Subgradient  Methods , Notes for EE364b, Stanford University, 2006. Available online.
      2. N. Shor, Minimization  Methods  for  Non-Differentiable  Functions , Springer Series in Computational Mathematics, Springer, 1985.
      3. D. Bertsekas, Nonlinear Programming. Athena Scientific, 1999.
    12. Robust Optimization
      1. Bertsimas, Dimitris, David B. Brown, and Constantine Caramanis. "Theory and applications of robust optimization." SIAM review 53.3 (2011): 464-501.
    13. Robust PCA
      1. Candes, Emmanuel J., et al. "Robust principal component analysis?." Journal of the ACM (JACM) 58.3 (2011): 11. 
      2. Liu, Guangcan, et al. "Robust recovery of subspace structures by low-rank representation." IEEE Transactions on Pattern Analysis and Machine Intelligence 35.1 (2013): 171-184.
      3. Wright, John, et al. "Robust principal component analysis: Exact recovery of corrupted low-rank matrices via convex optimization." Advances in neural information processing systems. 2009.   
      4. Xu, Huan, Constantine Caramanis, and Sujay Sanghavi. "Robust PCA via outlier pursuit." Advances in Neural Information Processing Systems. 2010.
    14. Matrix Completion
      1. Cai, Jian-Feng, Emmanuel J. Candes, and Zuowei Shen. "A singular value thresholding algorithm for matrix completion." SIAM Journal on Optimization 20.4 (2010): 1956-1982.       
      2. Candes, Emmanuel J., and Benjamin Recht. "Exact matrix completion via convex optimization." Foundations of Computational mathematics 9.6 (2009): 717.       
      3. Lin, Zhouchen, Minming Chen, and Yi Ma. "The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices." arXiv preprint arXiv:1009.5055 (2010).       
      4. Candes, Emmanuel J., and Terence Tao. "The power of convex relaxation: Near-optimal matrix completion." IEEE Transactions on Information Theory 56.5 (2010): 2053-2080.
    15. Coordinate ascent/descent
      1. Yu. Nesterov, Efficiency of coordinate descent methods on huge-scale optimization problems, SIAM Journal on Optimization, 2012
      2. P.  Richt ́arik and M. Takac, Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function, Mathematical Programming, 2014
      3. P. Tseng and S. Yun, A coordinate gradient descent method for nonsmooth separable minimization, Mathematical Programming, 2009
    16. Large-scale optimization
      1. Feng Niu, Benjamin Recht, Christopher Re, and Stephen Wright. Hogwild: A lock-free approach to parallelizing stochastic gradient descent. In Advances in neural information processing systems, pages 693–701, 2011
      2. Gemulla, Rainer, et al. "Large-scale matrix factorization with distributed stochastic gradient descent." Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining. ACM, 2011.       
      3. Bottou, Léon. "Large-scale machine learning with stochastic gradient descent." Proceedings of COMPSTAT'2010. Physica-Verlag HD, 2010. 177-186.
    17. Semi-stochastic methods:
      1. Konecny, Jakub, and Peter Richtarik. "Semi-stochastic gradient descent methods." arXiv preprint arXiv:1312.1666 (2013).
      2. Konecny, Jakub, et al. "Mini-batch semi-stochastic gradient descent in the proximal setting." IEEE Journal of Selected Topics in Signal Processing 10.2 (2016): 242-255.
      3. Konecny, Jakub, Zheng Qu, and Peter Richtarik. "Semi-stochastic coordinate descent." Optimization Methods and Software 32.5 (2017): 993-1005.
      4. Konecny, Jakub, Zheng Qu, and Peter Richtarik. "S2cd: Semi-stochastic coordinate descent." NIPS Optimization in Machine Learning workshop. 2014.
    18. Stochastic and online optimization
      1. A. Nemirovski, A. Juditsky, G. Lan and A. Shapiro,Robust stochastic approximation approach to stochastic programming, SIAM Journal on Optimization (2009)
      2. Yu. Nesterov, Primal-dual subgradient methods for convex problems, Mathematical Programming (2009)
      3. L. Xiao, Dual averaging methods for regularized stochastic learning and online optimization, Journal of Machine Learning Research (2010)
      4. N. Le Roux, M. Schmidt and F. Bach, A stochastic gradient method with an exponential convergence rate for strongly convex optimization with finite training sets, NIPS (2012)
    19. Variance reduced stochastic gradient descent
      1. R. Johnson and T. Zhang, Accelerating stochastic gradient descent using predictive variance reduction NIPS (2013)
      2. L. Xiao and T. Zhang, A proximal stochastic gradient method with progressive variance reduction
      3. Defazio, Aaron, Francis Bach, and Simon Lacoste-Julien. "SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives." Advances in neural information processing systems. 2014.
      4. Schmidt, Mark, Nicolas Le Roux, and Francis Bach. "Minimizing finite sums with the stochastic average gradient." Mathematical Programming 162.1-2 (2017): 83-112.
    20. Interior point methods and self-concordant analysis
      1. S. Boyd and L. Vandenberghe, Convex Optimization (2004), Chapter 9
    21. Proximal point algorithms:
      1. O. G ̈uler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control and Optimization (1991)
      2. O. G ̈uler,New proximal point algorithms for convex minimization, SIOPT (1992)
      3. O. G ̈uler, Augmented Lagrangian algorithm for linear programming, JOTA (1992)
    22. ADMM variants
      1. D. Goldfarb, S. Ma, K. Scheinberg, Fast alternating linearization methods for minimizing the sum of two convex functions, (2010)
      2. T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imag. Sciences (2009)
    23. Dual decomposition:
      1. F. Kelly, A. Maulloo, D. Tan, Rate control in communication networks: shadow prices, proportional fairness and stability, J. Operation Research Society, 49 (1998).
      2. A. Beck and M. Teboulle, Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Transactions on Image Processing (2009)
    24. Subgradient methods
      1. Nedic, Angelia, and Asuman Ozdaglar. "Distributed subgradient methods for multi-agent optimization." IEEE Transactions on Automatic Control 54.1 (2009): 48-61.
      2. Ram, S. Sundhar, Angelia Nedić, and Venugopal V. Veeravalli. "Distributed stochastic subgradient projection algorithms for convex optimization." Journal of optimization theory and applications 147.3 (2010): 516-545.
      3. Tsitsiklis, John, Dimitri Bertsekas, and Michael Athans. "Distributed asynchronous deterministic and stochastic gradient optimization algorithms." IEEE transactions on automatic control 31.9 (1986): 803-812.
    25. Smoothing
      1. Yu. Nesterov, Smooth minimization of non-smooth functions, Mathematical Programming (2005).
      2. Yu. Nesterov, Excessive gap technique in nonsmooth convex minimization, SIAM Journal on Optimization (2005)
    26. Majorization Minimization    
      1. Sun, Ying, Prabhu Babu, and Daniel P. Palomar. "Majorization-minimization algorithms in signal processing, communications, and machine learning." IEEE Transactions on Signal Processing 65.3 (2017): 794-816.       
      2. Figueiredo, Mario AT, Jose M. Bioucas-Dias, and Robert D. Nowak. "Majorization–minimization algorithms for wavelet-based image restoration." IEEE Transactions on Image processing 16.12 (2007): 2980-2991.       
      3. Mairal, Julien. "Incremental majorization-minimization optimization with application to large-scale machine learning." SIAM Journal on Optimization 25.2 (2015): 829-855. 
    27. Convex optimization in Finance
      1. Yiyong Feng and Daniel P. Palomar,  A Signal Processing Perspective on Financial Engineering, Foundations and Trends in Signal Processing, Now Publishers, vol. 9, no. 1-2, 2016.
    28. Blind deconvolution (Ken Ma
      1. Ahmed, Ali, Benjamin Recht, and Justin Romberg. "Blind deconvolution using convex programming." IEEE Transactions on Information Theory 60.3 (2014): 1711-1732.
      2. Chan, Tsung-Han, et al. "A convex analysis framework for blind separation of non-negative sources." IEEE Transactions on Signal Processing 56.10 (2008): 5120-5134.
      3. Gillis, Nicolas, and Stephen A. Vavasis. "Fast and robust recursive algorithmsfor separable nonnegative matrix factorization." IEEE transactions on pattern analysis and machine intelligence 36.4 (2014): 698-714.
    29. Network utility maximization (Palomar Chiang)       
      1. Palomar, Daniel Perez, and Mung Chiang. "A tutorial on decomposition methods for network utility maximization." IEEE Journal on Selected Areas in Communications 24.8 (2006): 1439-1451.       
      2. Palomar, Daniel P., and Mung Chiang. "Alternative distributed algorithms for network utility maximization: Framework and applications." IEEE Transactions on Automatic Control 52.12 (2007): 2254-2269.
    30. Congestion control
      1. Steven  H.  Low  and  David  E.  Lapsley,   “Optimization  Flow  Control,   I:  Basic Algorithm and Convergence,” IEEE/ACM Trans. on Networking, vol.  7, no.  6, Dec.  1999.
      2. Steven  H.  Low,  Larry  L.  Peterson,  and  Liming  Wang,  “Understanding  Vegas:  a duality model,” Journal of the ACM, vol.  49, no.  2, March 2002.
      3. Steven  H.  Low,  Fernando  Paganini,  and  John  C.  Doyle,  “Internet  Congestion Control,”  IEEE Control Systems Magazine, Feb.  2002.
      4. Steven H. Low, “A Duality Model of TCP and Queue Management Algorithms,” IEEE/ACM Trans. on Networking, Oct.  2003.
    31. Segmentation and multiview 3D reconstruction in image processing
      1. Daniel Cremers, Thomas Pock, Kalin Kolev, and Antonin Chambolle, “Convex Relaxation Techniques for Segmentation, Stereo and Multiview Reconstruction” In Markov Random Fields for Vision and Image Processing. MIT Press, 2011.
      2. Antonin Chambolle, Daniel Cremers, and Thomas Pock,“A convex approach to minimal partitions” SIAM Journal on Imaging Sciences. 5(4):1113–1158, 2012
      3. Kalin Kolev, Maria Klodt, Thomas Brox, and Daniel Cremers,“Continuous global optimization in multiview 3d reconstruction,” International Journal of Computer Vision. 84(1):80–96, 2009.
    32. MIMO linear transceiver design based on Schur convexity
      1. Daniel P. Palomar, “A Unified Framework for Communications through  MIMO  Channels,” Ph.D. dissertation, Technical University of Catalonia, Barcelona, Spain, May 2003.
      2. Daniel P. Palomar, “Convex Primal Decomposition for Multicarrier Linear MIMO Transceivers,” IEEE Trans. on Signal Processing, Vol. 53, No. 12, pp. 4661-4674, Dec. 2005.
      3. Daniel P. Palomar and Javier Fonollosa, “Algorithms for a Family of Waterfilling Solutions,” IEEE Trans. on Signal Processing, Vol. 53, No. 2, pp. 686-695, Feb. 2005.
    33. Iterative Waterfilling
      1. W. Yu, W. Rhee, S. Boyd, and J. Cioffi, “Iterative Water-Filling for Gaussian Vector Multiple-Access Channels,” IEEE Trans.  on Information Theory , vol. 50, no. 1, Jan. 2004.
      2. W. Yu, “A Dual Decomposition Approach to the Sum Power Gaussian Vector Multiple Access Channel Sum Capacity Problem,” in Proc. Conf. on Information and Systems (CISS), The Johns Hopkins Univ., March 12-14, 2003.
      3. N. Jindal, W. Rhee, S. Vishwanath, S. A. Jafar, and A. Goldsmith, “Sum Power Iterative Water-Filling for Multi-Antenna Gaussian Broad cast Channels,” IEEE Trans. on Information Theory , vol. 51, no. 4, April 2005.
      4. Daniel P. Palomar, “Convex Primal Decomposition for Multicarrier Linear MIMO Transceivers,” IEEE Trans.  on Signal Processing , vol. 53, no. 12, Dec. 2005.
    34. Minimax approaches in MIMO
      1. Daniel P. Palomar, John M. Cioffi, and Miguel A. Lagunas, “Uniform Power Allocation in MIMO Channels: A Game-Theoretic Approach,” IEEE Trans.  on Information Theory, vol. 49, no. 7, July 2003.
      2. Antonio Pascual-Iserte, Daniel P. Palomar, Ana P ́erez-Neira, Miguel A. Lagunas, “A Robust Maximim Approach for MIMO Comm.  with Imperfect CSI Based on Convex Optimization,” IEEE Trans. on Signal Processing, vol. 54, no. 1, Jan. 2006.
      3. Jiaheng Wang and Daniel P. Palomar, “Worst-Case Robust Transmission in MIMO Channels with Imperfect CSIT,” IEEE Trans.  on Signal Processing , vol.  57, no.  8, pp.  3086-3100, Aug. 2009.
    35. Game theoretic communication system design
      1. Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Competitive Design of Multiuser MIMO Systems based on Game Theory: A Unified View,” IEEE JSAC: Special Issue on Game Theory , vol. 25, no. 7, Sept. 2008.
      2. Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Optimal Linear Precoding Strategies for Wideband Noncooperative Systems Based on Game Theory,” IEEE Trans.  on Signal Processing , vol. 56, no. 3, March 2008.
      3. Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa , “Asynchronous Iterative Water-Filling for Gaussian Frequency-Selective Interference Channels,” IEEE Trans.  on Information Theory, vol. 54, no. 7, July 2008.
      4. Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa, “Cognitive MIMO Radio:  A Competitive Optimality Design Based on Subspace Projections,” IEEE Signal Processing Magazine, Nov. 2008.
    36. Signomial Programming:
      1. Mung  Chiang,  “Geometric  programming  for  communication  systems,” Foundations and Trends in Communications and Information Theory , Now Publishers, vol.  2, no.  1-2, Aug.  2006.
      2. Mung  Chiang,  Chee  Wei  Tan,  Daniel  P.  Palomar,  Daniel  O’Neill,  and  David Julian,  “Power Control by Geometric  Programming,” IEEE Trans.  on Wireless Communications, vol.  6, no.  7, pp.  2640-2651, July 2007
    37. Index tracking in finance
      1. Benidis, K., Feng, Y., and Palomar, D. P. (2017). Sparse Portfolios for High-Dimensional Financial Index Tracking. IEEE Transactions on Signal Processing
      2. Xu, F., Xu, Z., and Xue, H., (2015). Sparse index tracking based on L1/L2 model and algorithm. arXiv preprint
      3. Jansen, R., and Van Dijk, R. (2002). Optimal benchmark tracking with small portfolios The Journal of Portfolio Management vol. 28, no. 2, pp. 33–39.
      4. Beasley, J. E., Meade, N., and Chang, T.-J. (2003). An evolutionary heuristic for the index tracking problem. European Journal of Operational Research, vol. 148, no. 3, pp. 621–643.
      5. Scozzari, A., Tardella, F., Paterlini, S., and, Krink, T. (2013). Exact and heuristic approaches for the index tracking problem with UCITS constraints. Annals of Operations Research, vol. 205, no. 1, pp. 235–250.
    38. Multiobjective optimization and pareto optimality
      1. Marler, R. Timothy, and Jasbir S. Arora. "Survey of multi-objective optimization methods for engineering." Structural and multidisciplinary optimization 26.6 (2004): 369-395.
      2. Desideri, Jean-Antoine. "Multiple-gradient descent algorithm (MGDA) for multiobjective optimization." Comptes Rendus Mathematique 350.5-6 (2012): 313-318.       
      3. Fliege, Jorg, and Benar Fux Svaiter. "Steepest descent methods for multicriteria optimization." Mathematical Methods of Operations Research 51.3 (2000): 479-494.
      4. Harada, Ken, Jun Sakuma, and Shigenobu Kobayashi. "Local search for multiobjective function optimization: pareto descent method." Proceedings of the 8th annual conference on Genetic and evolutionary computation. ACM, 2006.
      5. Fliege, Joerg, LM Grana Drummond, and Benar Fux Svaiter. "Newton's method for multiobjective optimization." SIAM Journal on Optimization 20.2 (2009): 602-626.
    39. Saddle point methods for convex optimization
      1. Zhu, Minghui, and Sonia Martínez. "On distributed convex optimization under inequality and equality constraints." IEEE Transactions on Automatic Control 57.1 (2012): 151-164.
      2. Chang, Tsung-Hui, Angelia Nedić, and Anna Scaglione. "Distributed constrained optimization by consensus-based primal-dual perturbation method." IEEE Transactions on Automatic Control 59.6 (2014): 1524-1538.
    40. Integer programming
      1. Park, Jaehyun, and Stephen Boyd. "A semidefinite programming method for integer convex quadratic minimization." Optimization Letters (2017): 1-20.       
      2. Marchand, Hugues; Martin, Alexander; Weismantel, Robert; Wolsey, Laurence (2002). "Cutting planes in integer and mixed integer programming" (PDF). Discrete Applied Mathematics. 123: 387–446.
      3. Castro, J.; Nasini, S.; Saldanha-da-Gama, F. (2017). "A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method"
      4. Bader, David A.; Hart, William E.; Phillips, Cynthia A. (2004). "Parallel Algorithm Design for Branch and Bound" (PDF). In Greenberg, H. J. Tutorials on Emerging Methodologies and Applications in Operations Research. Kluwer Academic Press. 
    41. Distributed Optimization
      1. A.  Nedic  and  A.  Ozdaglar.   Distributed  subgradient  methods for multi-agent optimization. IEEE Transactions on Automatic Control , 54(1):48–61, Jan. 2009.
      2. J. C. Duchi, A. Agarwal, and M. J. Wainwright. "Dual averaging for distributed optimization: Convergence analysis and network scaling," IEEE Transactions on Automatic Control, 57(3):592–606, Mar. 2012
      3. E.  Wei  and  A.  Ozdaglar.    Distributed  alternating  direction method  of  multipliers.   In 51st  IEEE  Annual  Conference  on Decision and Control, pages 5445–5450, Dec. 2012.
      4. J. F. C. Mota, J. M. F. Xavier, P. M. Q. Aguiar, and M. Puschel. D-ADMM: A communication-efficient distributed algorithm for separable optimization. IEEE Transactions on Signal Processing, 61(10):2718–2723, May 2013.
      5. W.  Shi,  Q.  Ling,  K  Yuan,  G  Wu,  and  W  Yin.   On  the  linear convergence of the admm in decentralized consensus optimization. IEEE Transactions on Signal Processing , 62(7):1750–1761, April 2014.
    42. Cross-layer optimization in networks:       
      1. Lin, Xiaojun, Ness B. Shroff, and Rayadurgam Srikant. "A tutorial on cross-layer optimization in wireless networks." IEEE Journal on Selected areas in Communications 24.8 (2006): 1452-1463.       
      2. Chiang, Mung, et al. "Layering as optimization decomposition: A mathematical theory of network architectures." Proceedings of the IEEE 95.1 (2007): 255-312.       
      3. Georgiadis, Leonidas, Michael J. Neely, and Leandros Tassiulas. "Resource allocation and cross-layer control in wireless networks." Foundations and Trends in Networking 1.1 (2006): 1-144.       
      4. Lin, Xiaojun, and Ness B. Shroff. "Joint rate control and scheduling in multihop wireless networks." Decision and Control, 2004. CDC. 43rd IEEE Conference on. Vol. 2. IEEE, 2004.
    43. Convex optimization in Control Theory
      1. Linear Matrix Inequalities in System and Control Theory Stephen Boyd, Laurent El Ghaoui, E. Feron, and V. Balakrishnan
      2. Linear Controller Design – Limits of Performance, Stephen Boyd and Craig Barratt
    44. Convex optimization in Smart Grid
      1. Samadi, Pedram, et al. "Optimal real-time pricing algorithm based on utility maximization for smart grid." Smart Grid Communications (SmartGridComm), 2010 First IEEE International Conference on. IEEE, 2010.       
      2. Mohsenian-Rad, Amir-Hamed, et al. "Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid." IEEE transactions on Smart Grid 1.3 (2010): 320-331.       
      3. Sortomme, Eric, et al. "Coordinated charging of plug-in hybrid electric vehicles to minimize distribution system losses." IEEE transactions on smart grid 2.1 (2011): 198-205.
    45. Convex optimization in robotics       
      1. Verscheure, Diederik, et al. "Time-optimal path tracking for robots: A convex optimization approach." IEEE Transactions on Automatic Control 54.10 (2009): 2318-2327.       
      2. Schulman, John, et al. "Finding Locally Optimal, Collision-Free Trajectories with Sequential Convex Optimization." Robotics: science and systems. Vol. 9. No. 1. 2013.
      3. Zhu, Minghui, and Sonia Martinez. "On distributed convex optimization under inequality and equality constraints." IEEE Transactions on Automatic Control 57.1 (2012): 151-164.
      4. Derenick, Jason C., and John R. Spletzer. "Convex optimization strategies for coordinating large-scale robot formations." IEEE Transactions on Robotics 23.6 (2007): 1252-1259.
    46. Convex optimization for motion planning
      1. Schulman, John, et al. "Finding Locally Optimal, Collision-Free Trajectories with Sequential Convex Optimization." Robotics: science and systems. Vol. 9. No. 1. 2013.
      2. Prajna, Stephen, Pablo A. Parrilo, and Anders Rantzer. "Nonlinear control synthesis by convex optimization." IEEE Transactions on Automatic Control 49.2 (2004): 310-314.
      3. Schulman, John, et al. "Motion planning with sequential convex optimization and convex collision checking." The International Journal of Robotics Research 33.9 (2014): 1251-1270