Convex Analysis and Optimization

By Dimitri P. Bertsekas, Angelia Nedić, Asuman E. Ozdaglar. 2003

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Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition (2016, background): “…analogous to the gradient vector in the smooth case, the negation of the quantity argmin s ∈ ∂F ( x ) s points in the direction of steepest descent [see Bertsekas et al., 2003, Section 8.4] We ﬁrst derive an alternative representation of D g ( x, L ) in terms of the so-called forward-backward…”
Exact Matrix Completion via Convex Optimization (2008, background): “ that Xis solution to (1.5) if there exists a dual vector (or Lagrange multiplier) 2Rj jsuch that R is a subgradient of the nuclear norm at the point X, which we denote by R 2@kXk (3.1) (see, e.g. [7]). Recall the denition of a subgradient of a convex function f: Rn 1 n 2 !R. We say that Y is a subg”
Network Topology and Communication-Computation Tradeoffs in Decentralized Optimization (2017, methodology): “ k uk2 2 2 h(uk) h(u) + 2L2: 10Under weaker assumptions on the stepsize than those of Theorem 7, namely k !0 and P 1 k=0 k = 1, one can show that liminf k!1h(x k) = h, where h = inf x2Rdh(x), see [1], [42], [43]. 14 By summing these inequalities over k= 0;1;:::T, re-arranging the terms, and dividi”
Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs (2014, methodology): “0)k2 +c2), then we obtain kuk2 kvk2: Deﬁne the set Vto be the level set of q, as follows: V= fzjq(z) q(0) + 2(krq(0)k2 + c2): Being the level-set of a strongly-convex function, the set Vis compact [2](see Proposition 2.3.1(b), page 93). From the preceding discussion we see that if v62V, then kukkvk.”
Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization (2011, background): “Lemma 2 If x i is an ε i -optimal solution to the proximal problem (2) in the sense of (4) , then there exists f i such that (cid:107) f i Proof We ﬁrst recall some properties of ε -subdiﬀerentials (see, e.g., [41, Section 4.3] for more details).”
The Dirichlet problem for the convex envelope (2010, background): “Since Cx is closed and convex, then by the Plane Separation Theorem [3] there is an affine function L such that L(y) > 0 and L 0 in D.”
Trading regret for efficiency: online convex optimization with long term constraints (2011, methodology): “Although, at each iteration, λt is projected onto the Rm+ , since K is a compact set and functions ft(x) and gi(x), i ∈ [m] are convex, the boundedness of the functions implies that the gradients are bounded (Bertsekas et al., 2003, Proposition 4.2.3).”
Automatic Creation of Semantically Rich 3D Building Models from Laser Scanner Data (2011, methodology): “We ﬁnd the optimal value for w using stochastic gradient ascent (Bertsekas et al. (2003)).”
Incremental Stochastic Subgradient Algorithms for Convex Optimization (2008, unknown): “4) (see, for example, [2]).”