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We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
A class of smoothing methods is proposed for solving mathematical programs with equimibrium constraints. We introduce new and very simple regularizations of the complementarity constraints. Some estimate distance to optimal solution and…
A novel inner approximation algorithm is proposed for dynamic optimization problems to ensure strict satisfaction of path constraints. Distinct from traditional methods relying on interval analysis, the proposed algorithm leverages the…
Constraint solvers are complex pieces of software which require many design decisions to be made by the implementer based on limited information. These decisions affect the performance of the finished solver significantly. Once a design…
Current state-of-the-art methods for solving discrete optimization problems are usually restricted to convex settings. In this paper, we propose a general approach based on cutting planes for solving nonlinear, possibly nonconvex, binary…
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and…
Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…
Recent advances in deep learning have allowed neural networks (NNs) to successfully replace traditional numerical solvers in many applications, thus enabling impressive computing gains. One such application is time domain simulation, which…
A Pseudo-Boolean (PB) constraint is a linear inequality constraint over Boolean literals. One of the popular, efficient ideas used to solve PB-problems (a set of PB-constraints) is to translate them to SAT instances (encodings) via, for…
Neural networks are a powerful class of non-linear functions. However, their black-box nature makes it difficult to explain their behaviour and certify their safety. Abstraction techniques address this challenge by transforming the neural…
Neural network pruning is a fruitful area of research with surging interest in high sparsity regimes. Benchmarking in this domain heavily relies on faithful representation of the sparsity of subnetworks, which has been traditionally…
We study nonconvex optimization landscapes for learning overcomplete representations, including learning (i) sparsely used overcomplete dictionaries and (ii) convolutional dictionaries, where these unsupervised learning problems find many…
Outer approximation methods have long been employed to tackle a variety of optimization problems, including linear programming, in the 1960s, and continue to be effective for solving variational inequalities, general convex problems, as…
Neural networks are increasingly used as fast surrogate models across various domains, but unconstrained predictions can violate physical, operational, or safety requirements. We propose SnareNet, a feasibility-controlled architecture to…
Nominal abstract syntax is a popular first-order technique for encoding, and reasoning about, abstract syntax involving binders. Many of its applications involve constraint solving. The most commonly used constraint solving algorithm over…
A new computationally simple method of imposing hard convex constraints on the neural network output values is proposed. The key idea behind the method is to map a vector of hidden parameters of the network to a point that is guaranteed to…
For many applications, utilizing DNNs (Deep Neural Networks) requires their implementation on a target architecture in an optimized manner concerning energy consumption, memory requirement, throughput, etc. DNN compression is used to reduce…
We study the complexity of constraint satisfaction problems involving global constraints, i.e., special-purpose constraints provided by a solver and represented implicitly by a parametrised algorithm. Such constraints are widely used;…
Let $\rm{Box}_n = \{x \in \mathbb{R}^n : 0 \leq x \leq e \}$, and let $\rm{QPB}_n$ denote the convex hull of $\{(1, x')'(1, x') : x \in \rm{Box}_n\}$. The quadratic programming problem $\min\{x'Q x + q'x : x \in \rm{Box}_n\}$ where $Q$ is…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…