Computing an approximation of the nondominated set of multi-objective mixed-integer nonlinear optimization problems

In practical applications, one often has not only one, but several objectives that need to be optimized simultaneously. What is more, modeling such real world problems usually involves using both, continuous and integer variables. This then results in multi-objective mixed-integer optimization problems, which are in focus of this paper. We present an approximation concept, called … Read more

Solving Nonconvex Optimization Problems using Outer Approximations of the Set-Copositive Cone

We consider the solution of nonconvex quadratic optimization problems using an outer approximation of the set-copositive cone that is iteratively strengthened with conic constraints and cutting planes. Our methodology utilizes an MILP-based oracle for a generalization of the copositive cone that considers additional linear equality constraints. In numerical testing we evaluate our algorithm on a … Read more

On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, … Read more

A Computational Study for Piecewise Linear Relaxations of Mixed-Integer Nonlinear Programs

Solving mixed-integer nonlinear problems by means of piecewise linear relaxations can be a reasonable alternative to the commonly used spatial branch-and-bound. These relaxations have been modeled by various mixed-integer models in recent decades. The idea is to exploit the availability of mature solvers for mixed-integer problems. In this work, we compare different reformulations in terms … Read more

A hybrid branch-and-bound and interior-point algorithm for stochastic mixed-integer nonlinear second-order cone programming

One of the chief attractions of stochastic mixed-integer second-order cone programming is its diverse applications, especially in engineering (Alzalg and Alioui, {\em IEEE Access}, 10:3522-3547, 2022). The linear and nonlinear versions of this class of optimization problems are still unsolved yet. In this paper, we develop a hybrid optimization algorithm coupling branch-and-bound and primal-dual interior-point … Read more

Information Complexity of Mixed-integer Convex Optimization

We investigate the information complexity of mixed-integer convex optimization under different types of oracles. We establish new lower bounds for the standard first-order oracle, improving upon the previous best known lower bound. This leaves only a lower order linear term (in the dimension) as the gap between the lower and upper bounds. This is derived … Read more

Switching Time Optimization for Binary Quantum Optimal Control

Quantum optimal control is a technique for controlling the evolution of a quantum system and has been applied to a wide range of problems in quantum physics. We study a binary quantum control optimization problem, where control decisions are binary-valued and the problem is solved in diverse quantum algorithms. In this paper, we utilize classical … Read more

Gain Confidence, Reduce Disappointment: A New Approach to Cross-Validation for Sparse Regression

Ridge regularized sparse linear regression involves selecting a subset of features that explains the relationship between a high-dimensional design matrix and an output vector in an interpretable manner. To select the sparsity and robustness of linear regressors, techniques like leave-one-out cross-validation are commonly used for hyperparameter tuning. However, cross-validation typically increases the cost of sparse … Read more

Distributionally Ambiguous Multistage Stochastic Integer and Disjunctive Programs: Applications to Sequential Two-player Interdiction Games

This paper studies the generalizations of multistage stochastic mixed-integer programs (MSIPs) with distributional ambiguity, namely distributionally risk-receptive and risk-averse multistage stochastic mixed-integer programs (denoted by DRR- and DRA-MSIPs). These modeling frameworks have applications in non-cooperative Stackelberg games involving two players, namely a leader and a follower, with uncertainty in the impact of the decisions made … Read more