Convex Optimization Overview (cnt’d)
نویسنده
چکیده
In a convex optimization problem, x ∈ R is a vector known as the optimization variable, f : R → R is a convex function that we want to minimize, and C ⊆ R is a convex set describing the set of feasible solutions. From a computational perspective, convex optimization problems are interesting in the sense that any locally optimal solution will always be guaranteed to be globally optimal. Over the last several decades, general purpose methods for solving convex optimization problems have become increasingly reliable and efficient. In these lecture notes, we continue our foray into the field of convex optimization. In particular, we explore a powerful concept in convex optimization theory known as Lagrange duality. We focus on the main intuitions and mechanics of Lagrange duality; in particular, we describe the concept of the Lagrangian, its relation to primal and dual problems, and the role of the Karush-Kuhn-Tucker (KKT) conditions in providing necessary and sufficient conditions for optimality of a convex optimization problem.
منابع مشابه
Convex Optimization-based Beamforming: From Receive to Transmit and Network Designs
In this article, an overview of advanced convex optimization approaches to multi-sensor beamforming is presented, and connections are drawn between different types of optimization-based beamformers that apply to a broad class of receive, transmit, and network beamformer design problems. It is demonstrated that convex optimization provides an indispensable set of tools for beamforming, enabling ...
متن کاملNetwork Beamforming Is a Rapidly Emerging Area That Belongs to the General Field of Cooperative
n this article, an overview of advanced convex optimization approaches to multisensor beamforming is presented, and connections are drawn between different types of optimization-based beamformers that apply to a broad class of receive, transmit, and network beamformer design problems. It is demonstrated that convex optimization provides an indispensable set of tools for beamforming, enabling ri...
متن کاملAlternatives for optimization in systems and control: convex and non-convex approaches
In this presentation, we will develop a short overview of main trends of optimization in systems and control, and from there outline some new perspectives emerging today. More specifically, we will focus on the current situation, where it is clear that convex and Linear Matrix Inequality (LMI) methods have become the most common option. However, because of its vast success, the convex approach ...
متن کاملConvex Optimization for Big Data
This article reviews recent advances in convex optimization algorithms for Big Data, which aim to reduce the computational, storage, and communications bottlenecks. We provide an overview of this emerging field, describe contemporary approximation techniques like first-order methods and randomization for scalability, and survey the important role of parallel and distributed computation. The new...
متن کاملMeetings with Lambert W and Other Special Functions in Optimization and Analysis
We remedy the under-appreciated role of the Lambert W function in convex analysis and optimization. We first discuss the role of little-known special functions in optimization and then illustrate the relevance of W in a series problem posed by Donald Knuth. We then provide a basic overview of the convex analysis of W and go on to explore its role in duality theory where it appears quite natural...
متن کامل