Normal decomposition systems unify many results from convex matrix analysis regarding functions that are invariant with respect to a group of transformations—particularly those matrix functions that are unitarily-invariant and the affiliated permutation-invariant “spectral functions” that depend only on eigenvalues. Spectral functions extend in a natural way to Euclidean Jordan algebras, and several authors have studied … Read more
We consider mixed integer bilevel linear optimization problems in which the decision variables of the lower-level (follower’s) problem are all binary. We propose a general modeling and solution framework motivated by the practical reality that in a Stackelberg game, the follower does not always solve their optimization problem to optimality. They may instead implement a … Read more
The concept of balance plays an important role in many combinatorial optimization problems. Yet there exist various ways of expressing balance, and it is not always obvious how best to achieve it. In this methodology-focused paper, we study three cases where its integration is deficient and analyze the causes of these inadequacies. We examine the … Read more
This paper addresses the vehicle routing problem with time windows and stochastic demands (VRPTWSD). The problem is modeled as a two-stage stochastic program with recourse, in which routes are designed in the first stage and executed in the second. A failure occurs if the load of the vehicle is insufficient to meet the observed demand … Read more
This paper addresses a real-life short-term rescheduling problem of helicopter flights from one onshore airport to several maritime units in the context of the oil industry. This is a complex and challenging problem to solve because of the particular characteristics observed in practice, such as pending flights transferred from previous days with different recovering priorities, … Read more
Least squares form one of the most prominent classes of optimization problems, with numerous applications in scientific computing and data fitting. When such formulations aim at modeling complex systems, the optimization process must account for nonlinear dynamics by incorporating constraints. In addition, these systems often incorporate a large number of variables, which increases the difficulty … Read more
This paper addresses supervised learning problems with structured sparsity, where subsets of model coefficients form distinct groups. We introduce a novel log-composite regularizer in a bi-criteria optimization problem together with a loss (e.g., least squares) in order to reconstruct the desired group sparsity structure. We develop an iteratively reweighted algorithm that solves the group LASSO … Read more
Motivated by modern regression applications, in this paper, we study the convexification of a class of convex optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultaneously consider the nonlinear non-separable objective, indicator variables, and combinatorial constraints. Specifically, we give … Read more
Chemotherapy is one of the primary modalities of cancer treatment. Chemotherapy drug administration is a complex problem that often requires expensive clinical trials to evaluate potential regimens. One way to alleviate this burden and better inform future trials is to build reliable models for drug administration. Previous chemotherapy optimization models have mainly relied on optimal … Read more
We provide three new proofs of the strong concavity of the dual function of some convex optimization problems. For problems with nonlinear constraints, we show that the the assumption of strong convexity of the objective cannot be weakened to convexity and that the assumption that the gradients of all constraints at the optimal solution are … Read more