Convex optimization

Summary

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, finance, statistics (optimal experimental design), and structural optimization, where the approximation concept has proven to be efficient. With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming. Definition A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible set is a

Official source

https://en.wikipedia.org/wiki/Convex_optimization

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We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in the non-convex case. Under Lipschitz assumptions on the Riemannian gradient and Hessian of the cost function, these methods find approximate first-order critical points faster than regular gradient descent. A randomized version also finds approximate second-order critical points. Both the algorithms and their analyses build extensively on existing work in the Euclidean case. The basic operation consists in running the Euclidean accelerated gradient descent method (appropriately safe-guarded against non-convexity) in the current tangent space, then moving back to the manifold and repeating. This requires lifting the cost function from the manifold to the tangent space, which can be done for example through the Riemannian exponential map. For this approach to succeed, the lifted cost function (called the pullback) must retain certain Lipschitz properties. As a contribution of independent interest, we prove precise claims to that effect, with explicit constants. Those claims are affected by the Riemannian curvature of the manifold, which in turn affects the worst-case complexity bounds for our optimization algorithms.

SPRINGER2022

Audio Steganography using Convex Demixing

Yann Mikaël Schoenenberger

In the first part, we first introduce steganography (in chapter 1) not in the usual context of information security, but as a method to piggyback data on top of some content. We then focus on audio steganography and propose a new steganographic scheme in chapter 2 as well as a model for the noisy, analog communication channel we are considering (in section 3.2). The method we use is based on signal mixing, the science of constructing vectors in a way that their sum can then be split back into the original components. This is presented in chapter 4. The data recovery, or demixing, relies on convex optimization (presented in chapter 5) and some further signal processing detailed in chapter 6. In the second part we present our proof of concept implementation and show the results of simulation runs that have been made in order to study the properties of the overall system (chapter 7). We study how the different parameters of the system and the communication channel affect the performance of the steganographic scheme.Finally, we draw conclusions (chapter 8) about the findings and suggest (in chapter 9) what next steps could be taken in order to further study this steganographic scheme.

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Circuit optimization and PCB design for suspended microchannel resonators

Sarah Bonaly

2021