Introduction to linear optimization

By Dimitris J. Bertsimas, John N. Tsitsiklis. 1997

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Simulation-assisted machine learning (2018, background): “It falls into a subclass of linear optimization models called network flows which are used in a wide range of applications including production scheduling and transportation logistics (Bertsimas and Tsitsiklis, 1997).”
Universally utility-maximizing privacy mechanisms (2008, background): “For the rest of this section, ﬁx a user with prior { p i } and a loss function l ( i, r ) that is monotone in | i − r for every i ∈ N .”
Learning with Submodular Functions: A Convex Optimization Perspective (2011, methodology): “ linear programs may be solved eﬃciently by a number of methods, such as the simplex method (which uses heavily the polyhedral aspect of the problem, see §7.10) and interior-point methods (see, e.g., [24] and §7.5). – Representations as convex quadratic programs: The function Ψ(w) is then of the for”
On the conditions used to prove oracle results for the Lasso (2009, background): “The (S, s)-irrepresentable condition is sufficient and essentially necessary for variable selection An important characterization of the solution β∗ can be derived from the KarushKuhn-Tucker (KKT) conditions which in our context involves subdifferential calculus: see [1].”
MULTI-CATEGORY SUPPORT VECTOR MACHINES, FEATURE SELECTION AND SOLUTION PATH (2006, unknown): “A general parametric linear program can be written as It can be shown that the optimal solution ^ x ( (cid:18) ) is piecewise linear in (cid:18) and can be determined by tracking the basis B , consisting of those indices of the nonzero variables in x , c.f., Bertsimas and Tsitsiklis (1997).”
Identification and estimation of marginal effects in nonlinear panel models (2008, methodology): “This is equivalent to running least squares in the model Yit = βkXit + γk + vit, (5) for individuals with Xi = Xk, and averaging β̂k over k weighted by the sample frequencies of Xk.”
MAP estimation via agreement on trees: message-passing and linear programming (2005, methodology): “Although the method may perform well in practice, it is no longer guaranteed to output the correct MAP assignment, and it is straightforward to demonstrate problems on which it speciﬁes an incorrect (i.e., non-optimal) assignment.”
Learning with a Wasserstein Loss (2015, background): “[17]), hence the dual optimal ↵ is a subgradient of the loss with respect to its first argument.”
Spectral Clustering for Speech Separation (2009, methodology): “…1, and thus the relaxed solution has to be projected back to the constraint set defined by (a), an operation we refer to as “rounding,” due to the similarity with the rounding performed after a linear programming relaxation of an integer programming problem (Bertsimas and Tsitsiklis, 1997).”
Large Scale Visual Recognition (2012, background): “Hence there exists at least one basic feasible solution xB ofH [100], for which there are |B| linearly independent constraints that are active.”
The adaptive and the thresholded Lasso for potentially misspecified models (and a lower bound for the Lasso) (2011, methodology): “An important characterization of the solution βweight can be derived from the Karush-Kuhn-Tucker (KKT) conditions (see [3]).”
A Multi-Period Market Design for Markets With Intertemporal Constraints (2018, background): “In addition, because π i is the Lagrangian multiplier for the profit maximization problem (5)–(7) based on Property 1, it follows by sensitivity analysis in [25] that the overall profit change, i.”
Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems (2006, background): “Using that α is a basic feasible solution in the sense of [1] it is not hard to verify that α is also a vertex of Q(A).”
Optimal Residential Load Control With Price Prediction in Real-Time Electricity Pricing Environments (2010, background): “Therefore, it can be solved efficiently by using linear programming techniques [33].”