The paper identifies classes of nonconvex optimization problems whose convex relaxations have optimal solutions which at the same time are global optimal solutions of the original nonconvex problems. Such a hidden convexity property was so far limited to quadratically constrained quadratic problems with one or two constraints. We extend it here to problems with some … Read more
This paper presents an outcome-space outer approximation algorithm for solving the problem of minimizing the product of two convex functions over a compact convex set in $\R^n$. The computational experiences are reported. The proposed algorithm is convergent. ArticleDownload View PDF
We consider a general class of second-order iterations for unconstrained optimization that includes regularization and trust-region variants of Newton’s method. For each method in this class, we exhibit a smooth, bounded-below objective function, whose gradient is globally Lipschitz continuous within an open convex set containing any iterates encountered and whose Hessian is $\alpha-$Holder continuous (for … Read more
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 … Read more
The low-rank matrix recovery (LMR) is a rank minimization problem subject to linear equality constraints, and it arises in many fields such as signal and image processing, statistics, computer vision, system identification and control. This class of optimization problems is $NP$-hard and a popular approach replaces the rank function with the nuclear norm of the … Read more
We show that the robust counterpart of a convex quadratic constraint with ellipsoidal implementation error is equivalent to a system of conic quadratic constraints. To prove this result we first derive a sharper result for the S-lemma in case the two matrices involved can be simultaneously diagonalized. This extension of the S-lemma may also be … Read more
The optimal design of electrical machines can be mathematically modeled as a mixed-integer nonlinear optimization problem. We present six variants of such a problem, and we show, through extensive computational experiments, that, even though they are mathematically equivalent, the differences in the formulations may have an impact on the numerical performances of a local optimization … Read more
Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of … Read more
We discuss a modification of the chained Rosenbrock function introduced by Nesterov, a polynomial of degree four of $n$ variables. Its only stationary point is the global minimizer with optimal value zero. An initial point is given such that any continuous piecewise linear descent path consists of at least an exponential number of $0.72 \cdot … Read more
We address direct multiple shooting based algorithms for nonlinear model predictive control, with a focus on problems with long prediction horizons. We describe different efficient multiple shooting variants with a computational effort that is only linear in the horizon length. Proposed techniques comprise structure exploiting linear algebra on the one hand, and approximation of derivative … Read more