We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or machine learning. The paper discusses general-purpose modeling techniques and … Read more
Convex hull pricing provides a potential solution for reducing out-of-market payments in wholesale electricity markets. This paper revisits the theoretical construct of convex hull pricing and explores its important but underappreciated formulation-dependence property. Namely, convex hull prices may change for different formulations of the same unit commitment problem. After a conceptual exposition of the property, … Read more
Recent work has shown that stochastically perturbed gradient methods can efficiently escape strict saddle points of smooth functions. We extend this body of work to nonsmooth optimization, by analyzing an inexact analogue of a stochastically perturbed gradient method applied to the Moreau envelope. The main conclusion is that a variety of algorithms for nonsmooth optimization … Read more
We study the static chance-constrained lot sizing problem, in which production decisions over a planning horizon are made before knowing random future demands, and the backlog and inventory variables are then determined by the demand realizations. The chance constraint imposes a service level constraint requiring that the probability that any backlogging is required should be … Read more
These are lecture notes on bilevel optimization. The class of bilevel optimization problems is formally introduced and motivated using examples from different fields. Afterward, the main focus is on how to solve linear and mixed-integer linear bilevel optimization problems. To this end, we first consider various single-level reformulations of bilevel optimization problems with linear or … Read more
We present and analyze a new generalized Frank-Wolfe method for the composite optimization problem (P): F*:= min_x f(Ax) + h(x), where f is a \theta-logarithmically-homogeneous self-concordant barrier and the function h has bounded domain but is possibly non-smooth. We show that our generalized Frank-Wolfe method requires O((Gap_0 + \theta + Var_h)\ln(\delta_0) + (\theta + Var_h)^2/\epsilon) … Read more
In this work, we extend the regularization framework from Kronqvist et al. (https://doi.org/10.1007/s10107-018-1356-3) by incorporating several new regularization functions and develop a regularized single-tree search method for solving convex mixed-integer nonlinear programming (MINLP) problems. We propose a set of regularization functions based on distance-metrics and Lagrangean approximations, used in the projection problem for finding new … Read more
Exponents and logarithms exist in many important applications such as logistic regression, maximum likelihood, relative entropy and so on. Since the exponential cone can be viewed as the epigraph of perspective of the natural exponential function or the hypograph of perspective of the natural logarithm function, many mixed-integer nonlinear convex programs involving exponential or logarithm … Read more
Propagation of linear constraints has become a crucial sub-routine in modern Mixed-Integer Programming (MIP) solvers. In practice, iterative algorithms with tolerance-based stopping criteria are used to avoid problems with slow or infinite convergence. However, these heuristic stopping criteria can pose difficulties for fairly comparing the efficiency of different implementations of iterative propagation algorithms in a … Read more
Determining optimal prices in non-convex markets remains an unsolved challenge. Non-convex costs are critical in electricity markets, as startup costs and minimum operating levels yield a non-convex optimal value function over demand levels. While past research largely focuses on the performance of different non-convex pricing frameworks in the short-run, we determine long-run adapted resource mixes … Read more