We consider the {\it composite optimization problems} under convex and nonconvex settings. For the convex case, the {\it locally Lipschitz} condition is imposed on the gradient of the differentiable convex term. The classical {\it proximal gradient method} will be studied with our novel {\it enhanced adaptive} stepsize selection. To obtain the convergence of the proposed … Read more
Optimization over the set of matrices \(X\) that satisfy \(X^\top B X = I_p\), referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as the canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods … Read more
The feasibility-seeking approach offers a systematic framework for managing and resolving intricate constraints in continuous problems, making it a promising avenue to explore in the context of floorplanning problems with increasingly heterogeneous constraints. The classic legality constraints can be expressed as the union of convex sets. However, conventional projection-based algorithms for feasibility-seeking do not guarantee … Read more
A derivation of so-called “soft-margin support vector machines with kernel” is presented along with elementary proofs that do not rely on concepts from functional analysis such as Mercer’s theorem or reproducing kernel Hilbert spaces which are frequently cited in this context. The analysis leads to new continuity properties of the kernel functions, in particular a … Read more
This work introduces solar, a collection of ten optimization problem instances for benchmarking blackbox optimization solvers. The instances present different design aspects of a concentrated solar power plant simulated by blackbox numerical models. The type of variables (discrete or continuous), dimensionality, and number and types of constraints (including hidden constraints) differ across instances. Some are deterministic, others are stochastic … Read more
We propose an algorithm for general nonlinear conic programming which does not require the knowledge of the full cone, but rather a simpler, more tractable, approximation of it. We prove that the algorithm satisfies a strong global convergence property in the sense that it generates a strong sequential optimality condition. In particular, a KKT point … Read more
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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
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
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