Numerical splitting methods for nonsmooth convex optimization problems

In this thesis, we develop and investigate numerical methods for solving nonsmooth convex optimization problems in real Hilbert spaces. We construct algorithms, such that they handle the terms in the objective function and constraints of the minimization problems separately, which makes these methods simpler to compute. In the first part of the thesis, we extend the well known AMA method from Tseng to the Proximal AMA algorithm by introducing variable metrics in the subproblems of the primal-dual algorithm. For a special choice of metrics, the subproblems become proximal steps. Thus, for objectives in a lot of important applications, such as signal and image processing, machine learning or statistics, the iteration process consists of expressions in closed form that are easy to calculate. In the further course of the thesis, we intensify the investigation on this algorithm by considering and studying a dynamical system. Through explicit time discretization of this system, we obtain Proximal AMA. We show the existence and uniqueness of strong global solutions of the dynamical system and prove that its trajectories converge to the primal-dual solution of the considered optimization problem. In the last part of this thesis, we minimize a sum of finitely many nonsmooth convex functions (each can be composed by a linear operator) over a nonempty, closed and convex set by smoothing these functions. We consider a stochastic algorithm in which we take gradient steps of the smoothed functions (which are proximal steps if we smooth by Moreau envelope), and use a mirror map to 'mirror'' the iterates onto the feasible set. In applications, we compare them to similar methods and discuss the advantages and practical usability of these new algorithms

Effect of a conventional energy-restricted modified diet with or without meal replacement on weight loss and cardiometabolic risk profile in overweight women

<p>Abstract</p> <p>Background</p> <p>Abdominal obesity, atherogenic dyslipidemia and hypertension are essential risk factors for cardiovascular diseases. Several studies showed favorable effects of weight loss in overweight subjects on cardiometabolic risk profile.</p> <p>Methods</p> <p>This open-label, randomized, controlled study investigated the effect of an energy-restricted modified diet with (MR) or without meal replacements for weight control (C) on weight loss, body composition and cardiometabolic risk profile in overweight women. Of 105 randomized participants, 87 were eligible for per protocol analysis. Anthropometric, clinical, blood, 24 h-urine parameters and dietary intake were assessed at baseline and after 12 weeks.</p> <p>Results</p> <p>Dietary intervention resulted in a significant weight loss in both groups (MR: -5.98 ± 2.82 kg; p < 0.001, C: -4.84 ± 3.54 kg; p < 0.001). However, the rate of responder (weight loss >5%) was higher in MR (77%) versus C group (50%) (p = 0.010). A significant reduction in waist circumference (WC) and body fat mass (BFM) was observed in both groups. Body cell mass (BCM) and lean body mass (LBM) decreased, while percentage of BCM of body weight increased in MR more than in C group. Systolic and diastolic blood pressure (BP) significantly decreased and to a similar extent in both groups. Total cholesterol (TC), LDL-C but also HDL-C declined significantly in both groups, while no change occurred in triglycerides.</p> <p>Conclusions</p> <p>Both dietary intervention strategies had a similar effect on weight loss and body fat distribution, but rate of responder was significantly higher in MR group. Systolic BP decreased to a similar extent in both groups. Cardiometabolic risk profile improved only partly in both groups.</p

Numerical splitting methods for nonsmooth convex optimization problems

In this thesis, we develop and investigate numerical methods for solving nonsmooth convex optimization problems in real Hilbert spaces. We construct algorithms, such that they handle the terms in the objective function and constraints of the minimization problems separately, which makes these methods simpler to compute. In the first part of the thesis, we extend the well known AMA method from Tseng to the Proximal AMA algorithm by introducing variable metrics in the subproblems of the primal-dual algorithm. For a special choice of metrics, the subproblems become proximal steps. Thus, for objectives in a lot of important applications, such as signal and image processing, machine learning or statistics, the iteration process consists of expressions in closed form that are easy to calculate. In the further course of the thesis, we intensify the investigation on this algorithm by considering and studying a dynamical system. Through explicit time discretization of this system, we obtain Proximal AMA. We show the existence and uniqueness of strong global solutions of the dynamical system and prove that its trajectories converge to the primal-dual solution of the considered optimization problem. In the last part of this thesis, we minimize a sum of finitely many nonsmooth convex functions (each can be composed by a linear operator) over a nonempty, closed and convex set by smoothing these functions. We consider a stochastic algorithm in which we take gradient steps of the smoothed functions (which are proximal steps if we smooth by Moreau envelope), and use a mirror map to 'mirror'' the iterates onto the feasible set. In applications, we compare them to similar methods and discuss the advantages and practical usability of these new algorithms

Numerical splitting methods for nonsmooth convex optimization problems

In this thesis, we develop and investigate numerical methods for solving nonsmooth convex optimization problems in real Hilbert spaces. We construct algorithms, such that they handle the terms in the objective function and constraints of the minimization problems separately, which makes these methods simpler to compute. In the first part of the thesis, we extend the well known AMA method from Tseng to the Proximal AMA algorithm by introducing variable metrics in the subproblems of the primal-dual algorithm. For a special choice of metrics, the subproblems become proximal steps. Thus, for objectives in a lot of important applications, such as signal and image processing, machine learning or statistics, the iteration process consists of expressions in closed form that are easy to calculate. In the further course of the thesis, we intensify the investigation on this algorithm by considering and studying a dynamical system. Through explicit time discretization of this system, we obtain Proximal AMA. We show the existence and uniqueness of strong global solutions of the dynamical system and prove that its trajectories converge to the primal-dual solution of the considered optimization problem. In the last part of this thesis, we minimize a sum of finitely many nonsmooth convex functions (each can be composed by a linear operator) over a nonempty, closed and convex set by smoothing these functions. We consider a stochastic algorithm in which we take gradient steps of the smoothed functions (which are proximal steps if we smooth by Moreau envelope), and use a mirror map to 'mirror'' the iterates onto the feasible set. In applications, we compare them to similar methods and discuss the advantages and practical usability of these new algorithms