Convex Optimization Chapter 1 Introduction. What, Why and How  What is convex optimization  Why study convex optimization  How to study convex optimization.

Slides:

Advertisements

Similar presentations

Solving IPs – Cutting Plane Algorithm General Idea: Begin by solving the LP relaxation of the IP problem. If the LP relaxation results in an integer solution,

Advertisements

Mathematical Optimization in Stata: LP and MILP

Optimization in Engineering Design Georgia Institute of Technology Systems Realization Laboratory 123 “True” Constrained Minimization.

Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery.

Convex Position Estimation in Wireless Sensor Networks

Lecture 8 – Nonlinear Programming Models Topics General formulations Local vs. global solutions Solution characteristics Convexity and convex programming.

Nonlinear Programming

Discrete Optimization Shi-Chung Chang. Discrete Optimization Lecture #1 Today: Reading Assignments 1.Chapter 1 and the Appendix of [Pas82] 2.Chapter 1.

1 Spreadsheet Modeling & Decision Analysis: A Practical Introduction to Management Science 3d edition by Cliff Ragsdale.

Spreadsheet Modeling & Decision Analysis A Practical Introduction to Management Science 6 th edition Cliff T. Ragsdale © 2011 Cengage Learning. All Rights.

1 Chapter 8: Linearization Methods for Constrained Problems Book Review Presented by Kartik Pandit July 23, 2010 ENGINEERING OPTIMIZATION Methods and Applications.

ENGR 351 Numerical Methods Instructor: Dr. L.R. Chevalier

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. © 2001 South-Western/Thomson Learning. 8-1 Introduction to Nonlinear Programming (NLP)

Supply Chain Design Problem Tuukka Puranen Postgraduate Seminar in Information Technology Wednesday, March 26, 2009.

Solution methods for NP-hard Discrete Optimization Problems.

Sparse Kernels Methods Steve Gunn.

CS541 Advanced Networking 1 Introduction to Optimization Neil Tang 2/23/2009.

Optimization of Linear Problems: Linear Programming (LP) © 2011 Daniel Kirschen and University of Washington 1.

LP formulation of Economic Dispatch

CEM 514 Modeling of Construction Operations The LP/IP hybrid method for construction time-cost trade-off analysis By Samir Abdallah Sulaiman.

Linear Programming Operations Research – Engineering and Math Management Sciences – Business Goals for this section  Modeling situations in a linear environment.

Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies,

1 Chapter 8 Nonlinear Programming with Constraints.

CS 8751 ML & KDDSupport Vector Machines1 Support Vector Machines (SVMs) Learning mechanism based on linear programming Chooses a separating plane based.

© J. Christopher Beck Lecture 5: Project Planning 2.

Business Analytics with Nonlinear Programming

Nonlinear Programming.  A nonlinear program (NLP) is similar to a linear program in that it is composed of an objective function, general constraints,

Introduction to Linear Programming BSAD 141 Dave Novak.

Analytic Placement. Layout Project:  Sending the RTL file: −Thursday, 27 Farvardin  Final deadline: −Tuesday, 22 Ordibehesht  New Project: −Soon 2.

Introduction This module about Process Optimization was produced under the Program for North American Mobility in Higher Education as part of the Process.

Models in I.E. Lectures Introduction to Optimization Models: Shortest Paths.

Chapter 24 – Multicriteria Capital Budgeting and Linear Programming u Linear programming is a mathematical procedure, usually carried out by computer software,

OR Chapter 1. Introduction  Ex : Diet Problem Daily requirements : energy(2000kcal), protein(55g), calcium(800mg) Food Serving size Energy (kcal)

Spreadsheet Modeling & Decision Analysis A Practical Introduction to Management Science 5 th edition Cliff T. Ragsdale.

Robust Optimization and Applications Laurent El Ghaoui IMA Tutorial, March 11, 2003.

Section 5.5 The Real Zeros of a Polynomial Function.

Operations Research The OR Process. What is OR? It is a Process It assists Decision Makers It has a set of Tools It is applicable in many Situations.

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Branch and Bound Algorithms Present by Tina Yang Qianmei Feng.

Curve Fitting Introduction Least-Squares Regression Linear Regression Polynomial Regression Multiple Linear Regression Today’s class Numerical Methods.

Nonlinear Programming In this handout Gradient Search for Multivariable Unconstrained Optimization KKT Conditions for Optimality of Constrained Optimization.

Optimization in Engineering Design 1 Introduction to Non-Linear Optimization.

Linear Program MAX C B X B + C NB X NB s.t. BX B + A NB X NB = b X B, X NB ≥ 0.

Operations Research Models and Methods Advanced Operations Research Reference: Operations Research Models and Methods, Operations Research Models and Methods,

OR Integer Programming ( 정수계획법 ). OR

Chapter 8 Systems of Linear Equations in Two Variables Section 8.3.

Linear Programming Chapter 1 Introduction.

Instructional Design Document Simplex Method - Optimization STAM Interactive Solutions.

Linear Programming Wyndor Glass Co. 3 plants 2 new products –Product 1: glass door with aluminum framing –Product 2: 4x6 foot wood frame window.

Regularized Least-Squares and Convex Optimization.

1 Chapter 6 Reformulation-Linearization Technique and Applications.

1 Chapter 5 Branch-and-bound Framework and Its Applications.

Computational Approach for Adjudging Feasibility of Acceptable Disturbance Rejection Vinay Kariwala and Sigurd Skogestad Department of Chemical Engineering.

Decision Support Systems

Problem 1 Demand Total There are 20 full time employees, each can produce 10.

Solver & Optimization Problems

Solving Nonlinear Systems

Object Matching Using a Locally Affine Invariant and Linear Programming Techniques - H. Li, X. Huang, L. He Ilchae Jung.

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization Presenter: Xia Li.

Support Vector Machines Introduction to Data Mining, 2nd Edition by

Class Notes 11.2 The Quadratic Formula.

Sensitivity Analysis and

Least Squares Fitting A mathematical procedure for finding the best-fitting curve to a given set of points by minimizing the sum of the squares of the.

Chapter 1 Introduction Copyright © 2013 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

A more complex LP problem

Part 4 Nonlinear Programming

Communication Networks

Solving a System of Linear Equations

Solution methods for NP-hard Discrete Optimization Problems

Linear Constrained Optimization

Presentation transcript:

Convex Optimization Chapter 1 Introduction

What, Why and How  What is convex optimization  Why study convex optimization  How to study convex optimization

What is Convex Optimization?

Mathematical Optimization Convex Optimization Least-squaresLP Nonlinear Optimization

Mathematical Optimization

Convex Optimization

Least-squares

Analytical Solution of Least-squares f 0 ( x ) = jj A x ¡ b jj 2 2 = ( A x ¡ b ) > ( A x ¡ b ) x = ( A > A ) ¡ 1 A > 0 ( x x = 2 A > ( A x ¡ b ) = 0

Linear Programming (LP)

Why Study Convex Optimization?

Mathematical Optimization Convex Optimization Least-squaresLP Solving Optimization Problems Nonlinear Optimization

Analytical solution Good algorithms and software High accuracy and high reliability Time complexity: Mathematical Optimization Convex Optimization Least-squares LP Nonlinear Optimization A mature technology!

No analytical solution Algorithms and software Reliable and efficient Time complexity: Mathematical Optimization Convex Optimization Least-squares LP Nonlinear Optimization Also a mature technology!

Mathematical Optimization Convex Optimization Nonlinear Optimization Almost a mature technology! Least-squares LP No analytical solution Algorithms and software Reliable and efficient Time complexity (roughly)

Mathematical Optimization Convex Optimization Nonlinear Optimization Far from a technology! (something to avoid) Least-squares LP Sadly, no effective methods to solve Only approaches with some compromise Local optimization: “more art than technology” Global optimization: greatly compromised efficiency Help from convex optimization 1) Initialization 2) Heuristics 3) Bounds

Why Study Convex Optimization If not, …… -- Section 1.3.2, p8, Convex Optimization there is little chance you can solve it.

How to Study Convex Optimization?

Two Directions  As potential users of convex optimization  As researchers developing convex programming algorithms

Recognizing least-squares problems  Straightforward: verify the objective to be a quadratic function the quadratic form is positive semidefinite  Standard techniques increase flexibility Weighted least-squares Regularized least-squares

Recognizing LP problems  Example: Sum of residuals approximation Chebyshev or minimax approximation t = max i j a > i x ¡ b i j t i = j r i j

Recognizing Convex Optimization Problems

An Example

8 f j 1 ; j 2 ; ¢¢¢ ; j 10 g P 10 k = 1 p j k · 1 2 P m j = 1 p j Adding linear constraints????? C 10 m

Summary From the book, we expect to learn  To recognize convex optimization problems  To formulate convex optimization problems  To (know what can) solve them!