On the global convergence of a general class of augmented Lagrangian methods

In [E. G. Birgin, R. Castillo and J. M. Martínez, Computational Optimization and Applications 31, pp. 31-55, 2005], a general class of safeguarded augmented Lagrangian methods is introduced which includes a large number of different methods from the literature. Besides a numerical comparison including 65 different methods, primal-dual global convergence to a KKT point is … Read more

Exploiting Sign Symmetries in Minimizing Sums of Rational Functions

This paper is devoted to the problem of minimizing a sum of rational functions over a basic semialgebraic set. We provide a hierarchy of sum of squares (SOS) relaxations that is dual to the generalized moment problem approach due to Bugarin, Henrion, and Lasserre. The investigation of the dual SOS aspect offers two benefits: 1) … Read more

Approaches to iterative algorithms for solving nonlinear equations with an application in tomographic absorption spectroscopy

In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with variables coupling, we consider a situation where straightforward translation to a fixed point problem is not possible because the operators that represent the … Read more

Strengthening Lasserre’s Hierarchy in Real and Complex Polynomial Optimization

We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre’s hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensive numerical experiments show that our strengthening technique can significantly improve … Read more

Model Construction for Convex-Constrained Derivative-Free Optimization

We develop a new approximation theory for linear and quadratic interpolation models, suitable for use in convex-constrained derivative-free optimization (DFO). Most existing model-based DFO methods for constrained problems assume the ability to construct sufficiently accurate approximations via interpolation, but the standard notions of accuracy (designed for unconstrained problems) may not be achievable by only sampling … Read more

A Unified Approach for Maximizing Continuous $\gamma$-weakly DR-submodular Functions

\(\) This paper presents a unified approach for maximizing continuous \(\gamma\)-weakly DR-submodular functions that encompasses a range of settings and oracle access types. Our approach includes a Frank-Wolfe type offline algorithm for both monotone and non-monotone functions, with different restrictions on the convex feasible region. We consider settings where the oracle provides access to either … Read more

Sparse Polynomial Optimization with Unbounded Sets

This paper considers sparse polynomial optimization with unbounded sets. When the problem possesses correlative sparsity, we propose a sparse homogenized Moment-SOS hierarchy with perturbations to solve it. The new hierarchy introduces one extra auxiliary variable for each variable clique according to the correlative sparsity pattern. Under the running intersection property, we prove that this hierarchy … Read more

Block Majorization Minimization with Extrapolation and Application to $\beta$-NMF

We propose a Block Majorization Minimization method with Extrapolation (BMMe) for solving a class of multi-convex optimization problems. The extrapolation parameters of BMMe are updated using a novel adaptive update rule. By showing that block majorization minimization can be reformulated as a block mirror descent method, with the Bregman divergence adaptively updated at each iteration, … Read more