Optimization techniques for tree-structured nonlinear problems

Robust model predictive control approaches and other applications lead to nonlinear optimization problems defined on (scenario) trees. We present structure-preserving Quasi-Newton update formulas as well as structured inertia correction techniques that allow to solve these problems by interior-point methods with specialized KKT solvers for tree-structured optimization problems. The same type of KKT solvers could be used in active-set based SQP methods. The viability of our approach is demonstrated by two robust control problems

Relations between Abs-Normal NLPs and MPCCs. Part 2: Weak Constraint Qualifications

This work continues an ongoing effort to compare non-smooth optimization problems in abs-normal form to Mathematical Programs with Complementarity Constraints (MPCCs). We study general Nonlinear Programs with equality and inequality constraints in abs-normal form, so-called Abs-Normal NLPs, and their relation to equivalent MPCC reformulations. We introduce the concepts of Abadie's and Guignard's kink qualification and prove relations to MPCC-ACQ and MPCC-GCQ for the counterpart MPCC formulations. Due to non-uniqueness of a specific slack reformulation suggested in [10], the relations are non-trivial. It turns out that constraint qualifications of Abadie type are preserved. We also prove the weaker result that equivalence of Guginard's (and Abadie's) constraint qualifications for all branch problems hold, while the question of GCQ preservation remains open. Finally, we introduce M-stationarity and B-stationarity concepts for abs-normal NLPs and prove first order optimality conditions corresponding to MPCC counterpart formulations

Relations between Abs-Normal NLPs and MPCCs. Part 1: Strong Constraint Qualifications

This work is part of an ongoing effort of comparing non-smooth optimization problems in abs-normal form to MPCCs. We study the general abs-normal NLP with equality and inequality constraints in relation to an equivalent MPCC reformulation. We show that kink qualifications and MPCC constraint qualifications of linear independence type and Mangasarian-Fromovitz type are equivalent. Then we consider strong stationarity concepts with first and second order optimality conditions, which again turn out to be equivalent for the two problem classes. Throughout we also consider specific slack reformulations suggested in [9], which preserve constraint qualifications of linear independence type but not of Mangasarian-Fromovitz type

Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior

Long-time principal geodesic analysis in director-based dynamics of hybrid mechanical systems

In this article, we investigate an extended version of principal geodesic analysis for the unit sphere S2 and the special orthogonal group SO(3). In contrast to prior work, we address the construction of long-time smooth lifts of possibly non-localized data across branches of the respective logarithm maps. To this end, we pay special attention to certain critical numerical aspects such as singularities and their consequences on the numerical accuracy. Moreover, we apply principal geodesic analysis to investigate the behavior of several mechanical systems that are very rich in dynamics. The examples chosen are computationally modeled by employing a director-based formulation for rigid and flexible mechanical systems. Such a formulation allows to investigate our algorithms in a direct manner while avoiding the introduction of additional sources of error that are unrelated to principal geodesic analysis. Finally, we test our numerical machinery with the examples and, at the same time, we gain deeper insight into their dynamical behavior.publishedVersio

Branching Exponents of Synthetic Vascular Trees under Different Optimality Principles

Objective: The branching behavior of vascular trees is often characterized using Murray's law. We investigate its validity using synthetic vascular trees generated under global optimization criteria. Methods: Our synthetic tree model does not incorporate Murray's law explicitly. Instead, we show that its validity depends on properties of the optimization model and investigate the effects of different physical constraints and optimization goals on the branching exponent that is now allowed to vary locally. In particular, we include variable blood viscosity due to the Fåhræus–Lindqvist effect and enforce an equal pressure drop between inflow and the micro-circulation. Using our global optimization framework, we generate vascular trees with over one million terminal vessels and compare them against a detailed corrosion cast of the portal venous tree of a human liver. Results: Murray's law is fulfilled when no additional constraints are enforced, indicating its validity in this setting. Variable blood viscosity or equal pressure drop lead to different optima but with the branching exponent inside the experimentally predicted range between 2.0 and 3.0. The validation against the corrosion cast shows good agreement from the portal vein down to the venules. Conclusion: Not enforcing Murray's law increases the predictive capabilities of synthetic vascular trees, and in addition reduces the computational cost. Significance: The ability to study optimal branching exponents across different scales can improve the functional assessment of organs

SABMIS: sparse approximation based blind multi-image steganography scheme

We hide grayscale secret images into a grayscale cover image, which is considered to be a challenging steganography problem. Our goal is to develop a steganography scheme with enhanced embedding capacity while preserving the visual quality of the stegoimage as well as the extracted secret image, and ensuring that the stego-image is resistant to steganographic attacks. The novel embedding rule of our scheme helps to hide secret image sparse coefficients into the oversampled cover image sparse coefficients in a staggered manner. The stego-image is constructed by using the Alternating Direction Method of Multipliers (ADMM) to solve the Least Absolute Shrinkage and Selection Operator (LASSO) formulation of the underlying minimization problem. Finally, the secret images are extracted from the constructed stego-image using the reverse of our embedding rule. Using these components together, to achieve the above mentioned competing goals, forms our most novel contribution. We term our scheme SABMIS (Sparse Approximation Blind Multi-Image Steganography). We perform extensive experiments on several standard images. By choosing the size of the length and the width of the secret images to be half of the length and the width of cover image, respectively, we obtain embedding capacities of 2 bpp (bits per pixel), 4 bpp, 6 bpp, and 8 bpp while embedding one, two, three, and four secret images, respectively. Our focus is on hiding multiple secret images. For the case of hiding two and three secret images, our embedding capacities are higher than all the embedding capacities obtained in the literature until now (3 times and 6 times than the existing best, respectively). For the case of hiding four secret images, although our capacity is slightly lower than one work (about 2/3rd), we do better on the other two goals (quality of stego-image & extracted secret image as well as resistance to steganographic attacks). For our experiments, there is very little deterioration in the quality of the stego-images as compared to their corresponding cover images. Like all other competing works, this is supported visually as well as over 30 dB of Peak Signal-to-Noise Ratio (PSNR) values. The good quality of the stego-images is further validated by multiple numerical measures. None of the existing works perform this exhaustive validation. When using SABMIS, the quality of the extracted secret images is almost same as that of the corresponding original secret images. This aspect is also not demonstrated in all competing literature. SABMIS further improves the security of the inherently steganographic attack resistant transform based schemes. Thus, it is one of the most secure schemes among the existing ones. Additionally, we demonstrate that SABMIS executes in few minutes, and show its application on the real-life problems of securely transmitting medical images over the internet

Computational optimization of gas compressor stations: MINLP models versus continuous reformulations

When considering cost-optimal operation of gas transport networks, compressor stations play the most important role. Proper modeling of these stations leads to nonconvex mixed-integer nonlinear optimization problems. In this article, we give an isothermal and stationary description of compressor stations, state MINLP and GDP models for operating a single station, and discuss several continuous reformulations of the problem. The applicability and relevance of different model formulations, especially of those without discrete variables, is demonstrated by a computational study on both academic examples and real-world instances. In addition, we provide preliminary computational results for an entire network.German Federal Ministry of Economics and Technolog

Algorithmisches Programmieren (Numerische Algorithmen mit C++)

Dieser Kurs führt in die Programmiersprache C++ ein. Es werden die Grundlagen von C++, Kontrollstrukturen, Zahldarstellungen und Datentypen, Funktionen, Zeiger, objekt-orientierte Programmierung, Operatoren und deren Überladung, bishin zu Grundlagen der Vererbung und Klassentemplates, behandelt. Dieses Skriptum ist durch langjährige Erfahrungen der Autoren im Rahmen der gleichnamigen Vorlesung an der Leibniz Universität Hannover entstanden

Damage Location in Mechanical Structures by Multi-Objective Pattern Search

We propose a multi-objective global pattern search algorithm for the task of locating and quantifying damage in flexible mechanical structures. This is achieved by identifying eigenfrequencies and eigenmodes from measurements and matching them against the results of a finite element simulation model, which leads to a nonsmooth nonlinear bi-objective parameter estimation problem. A derivative-free optimization algorithm is required since the problem is nonsmooth and also because complex mechanical simulation models are often solved using commercial black-box software. Moreover, the entire set of non-dominated solutions is of interest to practitioners. Most solution approaches published to date are based on meta-heuristics such as genetic algorithms. The proposed multi-objective pattern-search algorithm provides a mathematically well-founded alternative. It features a novel sorting procedure that reduces the complexity in our context. Test runs on two experimental structures with multiple damage scenarios are used to validate the approach. The results demonstrate that the proposed algorithm yields accurate damage locations and requires moderate computational resources. From the engineer's perspective it represents a promising tool for structural health monitoring