Incorporating History and Deviations in Forward--Backward Splitting

We propose a variation of the forward--backward splitting method for solving structured monotone inclusions. Our method integrates past iterates and two deviation vectors into the update equations. These deviation vectors bring flexibility to the algorithm and can be chosen arbitrarily as long as they together satisfy a norm condition. We present special cases where the deviation vectors, selected as predetermined linear combinations of previous iterates, always meet the norm condition. Notably, we introduce an algorithm employing a scalar parameter to interpolate between the conventional forward--backward splitting scheme and an accelerated O(1/n^2)-convergent forward--backward method that encompasses both the accelerated proximal point method and the Halpern iteration as special cases. The existing methods correspond to the two extremes of the allowed scalar parameter range. By choosing the interpolation scalar near the midpoint of the permissible range, our algorithm significantly outperforms these previously known methods when addressing a basic monotone inclusion problem stemming from minimax optimization

Automated tight Lyapunov analysis for first-order methods

We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components, ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions. We provide a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality that amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle-Pock method when the linear operator is the identity mapping

Circuit Analysis using Monotone+Skew Splitting

It is shown that the behavior of an $m$-port circuit of maximal monotone elements can be expressed as a zero of the sum of a maximal monotone operator containing the circuit elements, and a structured skew-symmetric linear operator representing the interconnection structure, together with a linear output transformation. The Condat-V\~u algorithm solves inclusion problems of this form, and may be used to solve for the periodic steady-state behavior, given a periodic excitation at each port, using an iteration in the space of periodic trajectories.Comment: Submitted to the 2023 European Control Conferenc

Miejsce leków mukolitycznych w leczeniu schorzeń górnych dróg oddechowych

W pracy przedstawiono mechanizm działania leków mukolitycznych. Zaprezentowano ich podział, przedstawiono dostępne preparaty handlowe. Omówiono znaczenie mukolityków w wybranych schorzeniach górnych dróg oddechowych

Zapalenia języka i inne wybrane jego zmiany o charakterze łagodnym

W pracy przedstawiono zapalenia języka z uwzględnieniem ich symptomatologii, etiologii i czynników predysponujących. Zaprezentowano podział zapaleń języka. Poruszono problem nawracającego aftowego zapalenia jamy ustnej i języka. Przedstawiono niektóre łagodne zmiany języka jak np. zmiany atroficzne języka, do których należą język geograficzny i romboidalne zapalenie grzbietu języka. Omówiono także inną łagodną zmianę języka - język bruzdowaty. Forum Medycyny Rodzinnej 2008, tom 2, nr 2, 127-13