Related papers: Building Formulations for Piecewise Linear Relaxat…
The benefits of cutting planes based on the perspective function are well known for many specific classes of mixed-integer nonlinear programs with on/off structures. However, we are not aware of any empirical studies that evaluate their…
Recently, we proposed a class of inequalities called lifted bilinear cover inequalities, which are second-order cone representable convex inequalities, and are valid for a set described by a separable bilinear constraint together with…
We propose an approach to trajectory optimization for piecewise polynomial systems based on the recently proposed graphs of convex sets framework. We instantiate the framework with a convex relaxation of optimal control based on occupation…
We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate…
Nonsmooth functions have been used to model discrete-continuous phenomena such as contact mechanics, and are also prevalent in neural network formulations via activation functions such as ReLU. At previous AD conferences, Griewank et al.…
In this paper, we consider the problem of piecewise affine abstraction of nonlinear systems, i.e., the overapproximation of its nonlinear dynamics by a pair of piecewise affine functions that "includes" the dynamical characteristics of the…
We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a novel pairwise IB-representable class of CDCs, CDCs…
In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the…
In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients…
Many combinatorial optimisation problems can be modelled as valued constraint satisfaction problems. In this paper, we present a polynomial-time algorithm solving the valued constraint satisfaction problem for a fixed number of variables…
An important problem in optimization is the construction of mixed-integer programming (MIP) formulations of disjunctive constraints that are both strong and small. Motivated by lower bounds on the number of integer variables that are…
We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map…
In this study, we approach the analysis of a degenerate nonlinear functional in one dimension, accommodating a degenerate weight $w$. Our investigation focuses on establishing an explicit relaxation formula for a functional exhibiting…
This paper introduces a discrete relaxation for the class of combinatorial optimization problems which can be described by a set partitioning formulation under packing constraints. We present two combinatorial relaxations based on computing…
Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual…
Mixed integer linear programming (MILP) has seen a sharp rise in use for engineering optimization applications in recent years. Even for initially non-linear problems, it is often the method of choice. Then, the non-linear functions have to…
We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[\ell,u]$, where $z$ is a binary indicator of $x\in[\ell,u]$ ($0 \leq \ell <u$), and $y$ "captures" $f(x)$, which is assumed to be convex…
In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the total least squares (TLS) problem. We derive the explicit expressions of the mixed and componentwise condition numbers…
In this paper, we study some bounds for nonconvex quadratically constrained quadratic programs. We propose two types of bounds for quadratically constrained quadratic programs, quadratic and cubic bounds. For quadratic bounds, we use affine…
Solving mixed-integer nonlinear programs (MINLPs) typically relies on constructing relaxations that are easier to tackle than the original problem. Recently, global parabolic (PARA) relaxations were introduced, featuring separable quadratic…