An important example of this is the minimum distance. Convex optimization maxmin with constant mathematics. Lecture notes convex analysis and optimization electrical. Minmax and maxmin identities are found for inner products on the boundaries of compact, convex sets whose interiors contain the origin. Convex conjugate functions, conjugacy theorem, support functions. Convex slides 2014 massachusetts institute of technology. A theorem giving conditions on f, w and z in order to guarantee the saddle point property. As a matter of fact, our working horse will be convex optimization. Minmax identities on boundaries of convex sets around the. So there are a few optimizations to add to the algorithm. By switching the order of the minimization and maximization above, we obtain. Logarithmic regret algorithms for online convex optimization.
Convex analysis, duality and optimization cmu school of. Fang, on the entropic regularization method for min max problems with applications, ncsu or report, journal of optimization theory and applications submitted. Optimality and stability in nonconvexnonconcave minmax. Min commonmax crossing duality, weak duality, constrained optimization and minimax, strong duality. The authors analyze the gap between the static problem and the kadaptability problem and give. Telecommunications network design and maxmin optimization. Convex optimization maxmin with constant stack exchange. Byrne department of mathematical sciences university of massachusetts lowell a first course in optimization. Minmax saddlepoint problems have broad applications in many tasks in machine learning, e. Lastiterate convergence rates for minmax optimization. Min max saddlepoint problems have broad applications in many tasks in machine learning, e.
This choice is motivated by nice geometry of convex sets, functions, and optimization problems computational tractability of convex optimization implying computational ef. This notion can be expressed by an appropriate lg 1. Convex, concave, strictly convex, and strongly convex functions first and second order characterizations of convex functions optimality conditions for convex problems 1 theory of convex. Arise in the context of duality, network optimization, machine learning. Bertsekas2 abstract we provide a unifying framework for the visualization and analysis of duality, and other issues in convex optimization. Therefore its optimum is obtained over the boundary of convx. Hybrid block successive approximation for onesided non. Min max identities on boundaries of convex sets around the origin joseph f. Algorithms and applications songtao lu, ioannis tsaknakis, mingyi hong and yongxin chen abstract the min max problem, also known as the saddle point problem, is a class of optimization problems in which we minimize and maximize two subsets of variables simultaneously. Zang, a smoothing out technique for minmax optimization, mathematical programming 19, 6177 1980.
Bental and nemirovski 1998, 1999, 2000 address robust solutions minmaxmaxmin objective by allowing the uncertainty sets for the data to be ellipsoids, and propose efficient algorithms to solve convex optimization problems under data uncertainty. When the objective is weakly convex in terms of min variable and concave in terms of the max variable, we. It is based on two simple optimization problems that are dual to each other. Grcar lawrence berkeley national laboratory mail stop 50a1148 one cyclotron road berkeley, ca 947208142 usa email. Stoltz 2011 also considers some alternative notions of regret. Minmax identities on boundaries of convex sets around the origin. Although convexconcave saddlepoint problems have been broadly studied with efficient algorithms and solid theories available, it remains a challenge to design provably efficient algorithms for. Try to reproduce the numerical results in that reference.
We prove that global linear lastiterate convergence rates are achievable for more general classes of convexconcave minmax optimization problems than had previously been shown. Thanks in advance for any ideas about solving the problem or converting this problem into a linear optimization problem. Here are some of the topics that we will touch upon. Basic minmax algorithm 3 optimization however only very simple games can have their entire search tree generated in a short time. Karushkuhntucker conditions ryan tibshirani convex optimization 10725. Min max min robust combinatorial optimization 3 bertsimas and caramanis 4 introduced the concept of kadaptability. Consequently, an operator ais positive semide nite if and only. First a word of caution, optimization comes with a price. The identities resemble the minimax theorem but they are di. A geometric view of conjugacy in convex optimization 1 dimitri p. This section provides a complete set of lecture notes from the course along with the schedule of lecture topics. These applications require solving an optimization problem of the form min 2 max 2a f.
A simple geometric framework for convex optimization and minimax theory1 by dimitri p. This tutorial coincides with the publication of the new book on convex. Min max optimization of convex indefinite function. Convex optimization lecture notes for ee 227bt draft, fall. Convergence to a saddle point for convexconcave functions has been studied for decades, while the last few years have seen a surge of interest in nonconvexnonconcave minmax optimization due to the rise of deep learning.
Minmax and robust polynomial optimization springerlink. Of course, many optimization problems are not convex, and it can be di. A theorem giving conditions on f, w and z in order to guarantee the saddle point property is called a minimax theorem. Minimax optimal algorithms for unconstrained linear optimization. However, it remains an intriguing research challenge how local optimal points are defined and which algorithm can converge to such points.
For most games this isnt possible, the universe would probably vanish. What are the dimensions of the pen built this way that has the largest area. In this paper, we focus on a framework referred to as min common max crossing mcmc for short. Convex optimization lecture notes for ee 227bt draft, fall 20. Bental and nemirovski 1998, 1999, 2000 address robust solutions min max max min objective by allowing the uncertainty sets for the data to be ellipsoids, and propose efficient algorithms to solve convex optimization problems under data uncertainty. Min commonmax crossing duality theorems, strong duality conditions, existence of dual optimal solutions, nonlinear farkas lemma pdf 1. If is a solution of the constrained optimization problem, then there exists such that is a. In section 2 we introduce the lexicographic max min or the max min fair mmf solution concept and summarize its major properties. Minmaxmin robust combinatorial optimization 5 in the special case where no uncertain constant c 0 is considered, the objective function max c2uc xis linear on any line through the origin. A regularization method for solving the finite convex minmax problem, siam journal on numerical analysis 27, 16211634 1990. Distributed bregmandistance algorithms for minmax optimization kunal srivastava, angelia nedi. Lecture notes massachusetts institute of technology.
Although convex concave saddlepoint problems have been broadly studied with efficient algorithms and solid theories available, it remains a challenge to design provably efficient algorithms for. Since the sum of convex functions is always convex, l is convex for each. Linear optimization with max function convex constraint. Someconvexoptimizationproblemsinmachinelearning 233 we proceed to give a few important examples of convex optimization problemsinmachinelearning. Min commonproblem max crossingproblem weakduality q. Pdf approximate maxmin resource sharing for structured. A dualitybased approach for distributed minmax optimization with. A first course in optimization faculty server contact. Solving minmax problems and linear semiinfinite programs. Generalities primary methodology for large scale problems. Generalized polyhedral approximations in convex optimization pdf 2. Constrained optimization pieter abbeel uc berkeley eecs optional boyd and vandenberghe, convex optimization, chapters 9 11. Optimization 1 a rancher wants to build a rectangular pen, using one side of her barn for one side of the pen, and using 100m of fencing for the other three sides.
Let us finish this minitutorial by some promised examples. Higgins, superlinearly convergent algorithm for min max problems, journal of optimization theory and applications 69, 407439 1991. Distributed bregmandistance algorithms for minmax optimization. In section 2 we introduce the lexicographic maxmin or the maxmin fair mmf solution concept and summarize its major properties. In this paper, we focus on a framework referred to as min commonmax crossing mcmc for short. The idea is to compute k secondstage policies hereandnow.
Minimax optimal algorithms for unconstrained linear. In mathematics, the maxmin inequality is as follows. Solving a class of nonconvex minmax games using iterative. Ozdaglar2 abstract we provide a simple unifying framework for the visualization and analysis of convex programming duality and minimax saddle point theory. Hybrid block successive approximation for onesided nonconvex. Pdf optimality and stability in nonconvexnonconcave min. When equality holds one says that f, w and z satisfies a strong maxmin property or a saddlepoint property.1492 1258 1201 462 734 429 1526 551 1495 1096 822 317 12 990 576 514 312 47 1514 65 260 506 907 1302 182 131 936 628 506 71 1361 449 1000 1013 401 367 1038 1400 1013 734