Related papers: On Convex Envelopes and Regularization of Non-Conv…
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…
This paper presents a regularized Newton method (RNM) with generalized regularization terms for unconstrained convex optimization problems. The generalized regularization includes quadratic, cubic, and elastic net regularizations as special…
This paper builds upon our prior formalisation of R^n in ACL2(r) by presenting a set of theorems for reasoning about convex functions. This is a demonstration of the higher-dimensional analytical reasoning possible in our metric space…
In this paper, we study a class of problems where the sum of truncated convex functions is minimized. In statistical applications, they are commonly encountered when $\ell_0$-penalized models are fitted and usually lead to NP-Hard…
In this work, we introduce a new class of non-convex functions, called implicit concave functions, which are compositions of a concave function with a continuously differentiable mapping. We analyze the properties of their minimization by…
Nonconvex optimization problems arise in many areas of computational science and engineering and are (approximately) solved by a variety of algorithms. Existing algorithms usually only have local convergence or subsequence convergence of…
Weight optimization of frame structures with continuous cross-section parametrization is a challenging non-convex problem that has traditionally been solved by local optimization techniques. Here, we exploit its inherent semi-algebraic…
Nonconvex functionals with spherical symmetry are studied. Existence of one and radial symmetry of all global minimizers is shown with an approach based on convex relaxation.
In this paper we study the regularity of the local minima of integral functionals: in particular, not convexity (quasi-convexity, policonvexity or rank one convexity) hypothesis will be made on the density, neither structure hypothesis nor…
This paper presents a novel convex optimization-based method for finding the globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems with non-convex constraints that restrict the…
This work investigates the geometry of a nonconvex reformulation of minimizing a general convex loss function $f(X)$ regularized by the matrix nuclear norm $\|X\|_*$. Nuclear-norm regularized matrix inverse problems are at the heart of many…
Here we develop a regularity theory for a polyconvex functional in $2\times2-$dimensional compressible finite elasticity. In particular, we consider energy minimizers/stationary points of the functional…
The majorization-minimization (MM) principle is an extremely general framework for deriving optimization algorithms. It includes the expectation-maximization (EM) algorithm, proximal gradient algorithm, concave-convex procedure, quadratic…
We introduce and study various algorithms for solving nonconvex minimization with inequality constraints, based on the construction of convex surrogate envelopes that majorize the objective and the constraints. In the case where the…
Nonconvex regularization has been popularly used in low-rank matrix learning. However, extending it for low-rank tensor learning is still computationally expensive. To address this problem, we develop an efficient solver for use with a…
Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify…
In this paper, nonconvex and nonsmooth models for compressed sensing (CS) and low rank matrix completion (MC) is studied. The problem is formulated as a nonconvex regularized leat square optimization problems, in which the l0-norm and the…
In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain…
We present an efficient framework for solving algebraically-constrained global non-convex polynomial optimization problems over subsets of the hypercube. We prove the existence of an equivalent nonlinear reformulation of such problems that…
Low-rank matrix recovery problems are inverse problems which naturally arise in various fields like signal processing, imaging and machine learning. They are non-convex and NP-hard in full generality. It is therefore a delicate problem to…