The problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure. This paper describes a general procedure in using the projected gradient method. It is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly. This gives rise to ...

In this work, we propose an inexact projected gradient-like method for solving smooth constrained vector optimization problems. In the unconstrained case, we retrieve the steepest descent method introduced by Graña Drummond and Svaiter. In the constrained setting, the method we present extends the exact one proposed by Graña Drummond and Iusem, since it admits relative errors on the search dire...

A new active set algorithm for minimizing quadratic functions with separable convex constraints is proposed by combining the conjugate gradient method with the projected gradient. It generalizes recently developed algorithms of quadratic programming constrained by simple bounds. A linear convergence rate in terms of the Hessian spectral condition number is proven. Numerical experiments, includi...

Conic optimization is the minimization of a differentiable convex objective function subject to conic constraints. We propose novel primal–dual first-order method for optimization, named proportional–integral projected gradient (PIPG). PIPG ensures that both gap and constraint violation converge zero at rate O(1/k), where k number iterations. If strongly convex, improves convergence O(1/k2). Fu...

We present an efficient spectral projected-gradient algorithm for optimization subject to a group `1-norm constraint. Our approach is based on a novel linear-time algorithm for Euclidean projection onto the `1and group `1-norm constraints. Numerical experiments on large data sets suggest that the proposed method is substantially more efficient and scalable than existing methods.

We extend PGD and its recovery guarantee [1] from one-dimensional spectrally sparse signal recovery to the multi-dimensional case. Assume the underlying multi-dimensional spectrally sparse signal is of model order r and total dimension N . We show that O(r log(N)) measurements are sufficient for PGD to achieve successful recovery with high probability provided the underlying signal satisfies so...

An efficient gradient-based method to solve the volume constrained topology optimization problems is presented. Each iterate of this algorithm is obtained by the projection of a Barzilai-Borwein step onto the feasible set consisting of box and one linear constraints (volume constraint). To ensure the global convergence, an adaptive nonmonotone line search is performed along the direction that i...