In a recent paper we introduced a trust-region method with variable norms for unconstrained minimization and we proved standard asymptotic convergence results. Here we will show that, with a simple modification with respect to the sufficient descent condition and replacing the trust-region approach with a suitable cubic regularization, the complexity of this method for finding … Read more
Sequential optimality conditions for constrained optimization are necessarily satisfied by local minimizers, independently of the fulfillment of constraint qualifications. These conditions support the employment of different stopping criteria for practical optimization algorithms. On the other hand, when an appropriate strict constraint qualification associated with some sequential optimality condition holds at a point that satisfies the … Read more
Optimization problems over permutation matrices appear widely in facility layout, chip design, scheduling, pattern recognition, computer vision, graph matching, etc. Since this problem is NP-hard due to the combinatorial nature of permutation matrices, we relax the variable to be the more tractable doubly stochastic matrices and add an $L_p$-norm ($0 < p < 1$) regularization ... Read more
Any piecewise smooth function that is specified by an evaluation procedures involving smooth elemental functions and piecewise linear functions like min and max can be represented in the so-called abs-normal form. By an extension of algorithmic, or automatic differentiation, one can then compute certain first and second order derivative vectors and matrices that represent a … Read more
In this note, we recall two solutions to alleviate the catastrophic cancellations that occur when comparing function values in descent algorithms. The automatic finite differencing approach (Dussault and Hamelin) was shown useful to trust region and line search variants. The main original contribution is to successfully adapt the line search strategy (Hager and Zhang) for … Read more
This article deals with constrained multi-objective optimization problems. The main purpose of the article is to investigate relationships between constrained and unconstrained multi-objective optimization problems. Under suitable assumptions (e.g., generalized convexity assumptions) we derive a characterization of the set of (strictly, weakly) efficient solutions of a constrained multi-objective optimization problem using characterizations of the sets … Read more
We consider the problem of solving linear systems of equations with limited- memory members of the restricted Broyden class and symmetric rank-one matrices. In this paper, we present various methods for solving these linear systems, and propose a new approach based on a practical implementation of the compact representation for the inverse of these limited-memory … Read more
Implied volatility expansions allow calibration of sophisticated volatility models. They provide an accurate fit and parametrization of implied volatility surfaces that is consistent with empirical observations. Fine-grained higher order expansions offer a better fit but pose the challenge of finding a robust, stable and computationally tractable calibration procedure due to a large number of market … Read more
Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of $O(\epsilon^{-3/2})$ proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an $\epsilon$-approximate first-order critical point can be obtained under significantly weaker assumptions. Moreover, the result is generalized to the case where high-order derivatives are … Read more
Optimal control problems (OCP) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, it results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we … Read more