Related papers: Interior-Point Vanishing Problem in Semidefinite R…
We improve the effectiveness of propagation- and linear-optimization-based neural network verification algorithms with a new tightened convex relaxation for ReLU neurons. Unlike previous single-neuron relaxations which focus only on the…
While semidefinite programming (SDP) problems are polynomially solvable in theory, it is often difficult to solve large SDP instances in practice. One technique to address this issue is to relax the global positive-semidefiniteness (PSD)…
The global Lipschitz constant of a neural network is related to robustness and generalization, yet unlike in many classical models, it is not plainly legible from the parameters. This has motivated sophisticated verification algorithms,…
Despite the numerous uses of semidefinite programming (SDP) and its universal solvability via interior point methods (IPMs), it is rarely applied to practical large-scale problems. This mainly owes to the computational cost of IPMs that…
As deep neural networks are increasingly being deployed in practice, their efficiency has become an important issue. While there are compression techniques for reducing the network's size, energy consumption and computational requirement,…
For a large class of optimization problems, namely those that can be expressed as finite-valued constraint satisfaction problems (VCSPs), we establish a dichotomy on the number of levels of the Lasserre hierarchy of semi-definite programs…
Despite their impressive performance on diverse tasks, neural networks fail catastrophically in the presence of adversarial inputs---imperceptibly but adversarially perturbed versions of natural inputs. We have witnessed an arms race…
It has been recently proven that the semidefinite programming (SDP) relaxation of the optimal power flow problem over radial networks is exact under technical conditions such as not including generation lower bounds or allowing load…
Tight estimation of the Lipschitz constant for deep neural networks (DNNs) is useful in many applications ranging from robustness certification of classifiers to stability analysis of closed-loop systems with reinforcement learning…
Correspondence problems are often modelled as quadratic optimization problems over permutations. Common scalable methods for approximating solutions of these NP-hard problems are the spectral relaxation for non-convex energies and the…
Deep Neural Networks (DNNs) are powerful tools that have shown extraordinary results in many scenarios, ranging from pattern recognition to complex robotic problems. However, their intricate designs and lack of transparency raise safety…
The trust-region problem, which minimizes a nonconvex quadratic function over a ball, is a key subproblem in trust-region methods for solving nonlinear optimization problems. It enjoys many attractive properties such as an exact…
Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist…
In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite…
We introduce Sieve-SDP, a simple facial reduction algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP inspects the constraints of the problem to detect lack of strict feasibility, deletes redundant rows and columns, and reduces…
Analyzing the worst-case performance of deep neural networks against input perturbations amounts to solving a large-scale non-convex optimization problem, for which several past works have proposed convex relaxations as a promising…
Motivated by applications in wireless communications, this paper develops semidefinite programming (SDP) relaxation techniques for some mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation…
We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing…
This thesis explores algorithmic applications and limitations of convex relaxation hierarchies for approximating some discrete and continuous optimization problems. - We show a dichotomy of approximability of constraint satisfaction…
Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically…