Convex Optimization Theory

By Dimitri P. Bertsekas. 2009

855 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/e1932c4db44cc2ceab12a347c6067b139d040abb

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Citations in the corpus

Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization (2019, result): “…equality (see, e.g., Ex 5.25 in Boyd & Vandenberghe (2004)) Together, the above results imply that ( θ ∗ , Q ∗ ) form a saddle point (proposition 3.4.1, Bertsekas (2009)), that is, In particular, the second equality indicates that the optimal DRO model θ ∗ also minimizes the weighted risk h (…”
Learning Student Networks with Few Data (2020, background): “(Mohajerin Esfahani and Kuhn 2018, Proposition 6.5) shows that if (cid:2) is L - lipschitz continuous, then Furthermore, the Lipschitz constant can be larger to the supremum module of the (sub)gradient over X (Bertsekas 2009), which completes the proof.”
Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems (2015, methodology): “The following concepts and properties about convex functions, convex sets, and projection operations come from Ruszczynski (2006) and Bertsekas (2009).”
Adversarial Robustness with Semi-Infinite Constrained Learning (2021, background): “Hence, strong duality holds [96, Ch. 4] and we obtain that p ( θ ) = max for the dual function d PI .”
User Association for Load Balancing in Heterogeneous Cellular Networks (2012, background): “e dynamically updates according to the rule (t) = (t) D((t)) D(t) k@D((t))k2 ; 0 < (t) < 2; (21) where D(t) is an estimate of the optimal value D of problem (15), and are some scalars [30]. We consider a procedure for updating D(t), whereby D(t) is given by D(t) = min 0˝t D((˝)) "(t); ”
Robust Energy Management for Microgrids With High-Penetration Renewables (2012, unknown): “The iterates are guaranteed to converge to a neighborhood of the optimal multipliers [25, Sec.”
Energy Efficient Resource Allocation in UAV-Enabled Mobile Edge Computing Networks (2019, unknown): “ because hij means the power consumption if UE iuploads data to UAV jaccording to (A.2). Note that the 12 value of {αi}i∈N,{βj}j∈M,{γi}i∈N,{λj}j∈M,{µj}j∈M can be determined by the sub-gradient method [53]. The updating procedure can be given by βi = " βi +φ X j∈Mi aijCij +ai0Ei − T !# + (34) γi”
Distributionally Robust Logistic Regression (2015, unknown): “∗(θ) −hq,ξˆ−ξi = sup 0≤θ≤1 inf kqk∗≤λ sup ξ∈Rn hθβ +q,ξi −f∗(θ) −hq,ξˆi, where the third equality follows from the deﬁnition of the du al norm, and the last equality holds due to Proposition 5.5.4 in [33]. Explicitly evaluating the maximization over ξ shows that the above expression is equivalent to”
Constrained Learning With Non-Convex Losses (2021, methodology): “In fact, there may not even be a set of weights (regularization parameters) that yields a solution of (P-ECRM), leading to infeasible results or unacceptably poor performance [33].”
Energy Efficient User Association and Power Allocation in Millimeter-Wave-Based Ultra Dense Networks With Energy Harvesting Base Stations (2017, unknown): “In updating the variable pij , we use Newton’s method which has Algorithm 1 Iterative Gradient Algorithm 1: Initialization: Imax and pmax, Let pij = p0 2: Initialization: Set the Lagrange multipliers μj , λj and νi to zero 3: Set i = 0 4: repeat 5: User association 6: for i = 1 to U do 7: for j = 1 ”
Diff-DAC: Distributed Actor-Critic for Multitask Deep Reinforcement Learning (2017, methodology): “Therefore, we can use Weierstrass’ Theorem (Bertsekas 2009, Prop.”
Regularization via Mass Transportation (2017, background): “If L represents the smooth hinge loss function and Corollary 3.14 (Logistic Regression) .”