We introduce a sequential optimality condition for locally Lipschitz constrained nonsmooth optimization, verifiable just using derivative information, and which holds even in the absence of any constraint qualification. The proposed sequential optimality condition is not only novel for nonsmooth problems, but brings new insights for the smooth case as well. We present a practical algorithm … Read more
In this paper, we study the loss surface of the over-parameterized fully connected deep neural networks. We prove that for any continuous activation functions, the loss function has no bad strict local minimum, both in the regular sense and in the sense of sets. This result holds for any convex and differentiable loss function, and … Read more
In this paper, we consider the nonconvex quadratic optimization problem with a single quadratic constraint. First we give a theoretical characterization of the local non-global minimizers. Then we extend the recent characterization of the global minimizer via a generalized eigenvalue problem to the local non-global minimizers. Finally, we use these results to derive an efficient … Read more
We consider feasibility problems defined by a set of constraints that exhibit gradient H\”older continuity plus additional constraints defined by the affordability of obtaining approximate minimizers of quadratic models onto the associated feasible set. Each iteration of the method introduced in this paper involves the approximate minimization of a two-norm regularized quadratic subject to the … Read more
The generalized problem of moments is a conic linear optimization problem over the convex cone of positive Borel measures with given support. It has a large variety of applications, including global optimization of polynomials and rational functions, options pricing in finance, constructing quadrature schemes for numerical integration, and distributionally robust optimization. A usual solution approach, … Read more
This paper is concerned with the zero-norm regularized quadratic optimization with a sphere constraint, which has an important application in sparse eigenvalue problems. For this class of nonconvex and nonsmooth optimization problems, we establish the KL property of exponent 1/2 for its extended-valued objective function and develop a globally and linearly convergent proximal gradient method … Read more
In this paper, based on a modified Gram-Schmidt (MGS) process, we propose a class of derivative-free conjugate gradient (CG) projection methods for nonsmooth equations with convex constraints. Two attractive features of the new class of methods are: (i) its generated direction contains a free vector, which can be set as any vector such that the … Read more
We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang’s asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73-88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop … Read more
A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is beta-H\”{o}lder continuous. It features a … Read more
Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, vari- able selection, and high-dimensional analysis. We propose a broad framework for sparse relaxed regularized regression, called SR3. The key idea is to solve a relaxation of … Read more