Convex Analysis and Optimization

By Arindam Banerjee. 2006

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Citations in the corpus

Stochastic Primal-Dual Q-Learning (2018, neutral): “…constraints, and consider the Lagrangian function Note that when setting M = I |S||A| in L M , denoted by L I , it reduces to the Lagrangian function (10) for (9), i.e., Then, the optimal solution can be obtained by solving the saddle point problem min According to Bertsekas et al. (2003, Prop.”
Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization (2008, background): “Thus, the subgradients of each fi are also bounded (see [35], Proposition 4.”
Distributed subgradient methods and quantization effects (2008, positive): “For a convex function F : R → R, we use the notion of a subgradient (see [2]): a vector sF (x̄) ∈ R is a subgradient of a convex function F at x̄ if F (x̄) + sF (x̄)′(x− x̄) ≤ F (x) for all x.”
On risk bounds in isotonic and other shape restricted regression problems (2013, unknown): “ction y →θˆ(y;K) is well deﬁned [because for each y and K, the quantity θˆ(y;K) exists uniquely by the Hilbert projection theorem], nonlinear in y (in general) and can be characterized by (see, e.g., [4], Proposition 2.2.1) (13) θˆ(y;K)∈K, hy−θˆ(y;K),θˆ(y;K)i=0 and hy−θˆ(y;K),ωi≤0 for all ω ∈K. Ineq”
Energy optimal control for time-varying wireless networks (2005, background): “Average power can be further reduced if channel states and arrivals are extended periodically (or probabilistically) over the infinite time horizon, and it can be shown that the minimum average power required to stabilize such a system is given by .”
MIMO Transceiver Design via Majorization Theory (2007, methodology): “A unified framework is developed that hinges on majorization theory as a key tool to transform the originally complicated matrix-valued nonconvex problem into a simple scalar problem in most cases convex which can be addressed under the powerful framework of convex optimization theory [12, 13, 20].”
Convex Optimization & Euclidean Distance Geometry (2004, methodology): “When C1 is any member of a convex set C whose dimension is L , Caratheodory’s theorem [57] [182] [114] [25] [226] guarantees that no more than L+1 affinely independent members from C are required to faithfully represent it by some linear combination of those members.”
Recovering Low-Rank Matrices From Few Coefficients in Any Basis (2009, background): “not just for generic matrices ρ+ ∆ in the aforementioned afﬁne plane, but for all such elements simultaneously. Key to that will be a simple concept from convex optimization theory: a dual certiﬁcate [23], [9], [2], [3]. By that we mean a matrix Y such that kρ+∆k 1 >kρk 1 +(Y,∆) (10) for ∆ 6= 0 .”
On Distributed Convex Optimization Under Inequality and Equality Constraints (2010, methodology): “This may cause the subgradient S ix ( k ) not to be uniformly bounded, while the boundedness of subgradients is a standard assumption in the analysis of subgradient methods, e.g., see [2], [3], [14], [15], [16], [17].”
Data-driven robust optimization (2013, background): “Note, the existence of the set in Step 3 is guaranteed by the bijection between closed, finite-valued, positively homogenous convex functions and convex, compact sets (see [11]).”
Convex Optimization, Game Theory, and Variational Inequality Theory (2010, background): “We refer, e.g., to [24] and [25] for details, but it is safe to say that, to a large extent, convex optimization problems constitute the largest class of tractable optimization problems.”
Distributed Asynchronous Constrained Stochastic Optimization (2011, methodology): “The consensus algorithm based on a broadcast based communication protocol was discussed in [26].”
A tutorial on decomposition methods for network utility maximization (2006, background): “value converges to within some range of the optimal value (assuming bounded subgradients) [12].”
Incremental Learning-to-Learn with Statistical Guarantees (2018, methodology): “(Bertsekas et al., 2003), and we can rely on the convergence of Alg.”