Basic Concepts of Optimization

Slides:



Advertisements
Similar presentations
What is Optimal Control Theory? Dynamic Systems: Evolving over time. Time: Discrete or continuous. Optimal way to control a dynamic system. Prerequisites:
Advertisements

1 OR II GSLM Outline  some terminology  differences between LP and NLP  basic questions in NLP  gradient and Hessian  quadratic form  contour,
Introducción a la Optimización de procesos químicos. Curso 2005/2006 BASIC CONCEPTS IN OPTIMIZATION: PART II: Continuous & Unconstrained Important concepts.
Geometry Chapter Polygons. Convex Polygon – a polygon with a line containing a side with a point in the interior of the polygon.
Polygons – Concave and Convex Turning Point Quiz Copyright © 2010 Kelly Mott.
Lecture 8 – Nonlinear Programming Models Topics General formulations Local vs. global solutions Solution characteristics Convexity and convex programming.
Frank Cowell: Microeconomics Convexity Microeconomia III (Lecture 0) Tratto da Cowell F. (2004), Principles of Microeoconomics November 2004.
D Nagesh Kumar, IIScOptimization Methods: M2L2 1 Optimization using Calculus Convexity and Concavity of Functions of One and Two Variables.
Economics 2301 Lecture 31 Univariate Optimization.
Nonlinear Programming
Frank Cowell: Convexity CONVEXITY MICROECONOMICS Principles and Analysis Frank Cowell 1 March 2012.
Agenda Duality (quickly) Piecewise linearity Start chapter 4.
©Brooks/Cole, 2001 Chapter 9 Pointers. ©Brooks/Cole, 2001 Figure 9-1.
OPTIMAL CONTROL SYSTEMS
Support Vector Machines Formulation  Solve the quadratic program for some : min s. t.,, denotes where or membership.  Different error functions and measures.
Economics 2301 Lecture 5 Introduction to Functions Cont.
©Brooks/Cole, 2001 Chapter 8 Arrays. ©Brooks/Cole, 2001 Figure 8-1.
Economics 214 Lecture 8 Introduction to Functions Cont.
©Brooks/Cole, 2001 Chapter 3 Structure of a C Program.
©Brooks/Cole, 2001 Chapter 10 Pointer Applications.
©Brooks/Cole, 2001 Chapter 11 Strings. ©Brooks/Cole, 2001 Figure 11-1.
Multivariable Optimization
D Nagesh Kumar, IIScOptimization Methods: M2L5 1 Optimization using Calculus Kuhn-Tucker Conditions.
Optimality Conditions for Nonlinear Optimization Ashish Goel Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A.
P247. Figure 9-1 p248 Figure 9-2 p251 p251 Figure 9-3 p253.
D Nagesh Kumar, IIScOptimization Methods: M2L3 1 Optimization using Calculus Optimization of Functions of Multiple Variables: Unconstrained Optimization.
Optimization Theory Primal Optimization Problem subject to: Primal Optimal Value:
©Brooks/Cole, 2001 Chapter 4 Functions. ©Brooks/Cole, 2001 Figure 4-1.
Trading Convexity for Scalability Marco A. Alvarez CS7680 Department of Computer Science Utah State University.
1 OR II GSLM Outline  separable programming  quadratic programming.
Introduction to Optimization (Part 1)
Warm Up: Investigating the Properties of Quadrilaterals Make a conjecture about the sum of the interior angles of quadrilaterals. You may use any material/equipment.
1. Problem Formulation. General Structure Objective Function: The objective function is usually formulated on the basis of economic criterion, e.g. profit,
Objectives In this chapter you will:  Find measures of interior and exterior angles of polygons  Solve problems involving angle measures of polygons.
AP/Honors Calculus Chapter 4 Applications of Derivatives Chapter 4 Applications of Derivatives.
Slide 6-1 Copyright © 2004 Pearson Education, Inc.
Polygon: Many sided figure Convex Convex vs. Nonconvex.
6.1 Polygons.
1.6 Classify Polygons. A polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. A polygon that.
L8 Optimal Design concepts pt D
Performance Surfaces.
Chapter 1-6 (Classify Polygons)  What is a polygon?  A closed plane figure formed by 3 or more line segments, with no two sides being collinear.
§10.1 Polygons  Definitions:  Polygon  A plane figure that…  Is formed by _________________________ called sides and… ..each side intersects ___________.
Continuous Optimization. Copyright by Yu-Chi Ho2 First and second N.A.S.C. F Unconstraint optimization problem F Performance index: F Decision vector:
Current and Resistance
L6 Optimal Design concepts pt B
deterministic operations research
Relative Extrema and More Analysis of Functions
Computational Optimization
Work – review from gr 11 Units - Joules (J) = Nm = kgm2s-2
GSP 321 Innovative Education-- snaptutorial.com
Lecture 8 – Nonlinear Programming Models
Applications of the Derivative
Notes #6 1-6 Two Dimensional Figures
Introduction and Mathematical Concepts
CHAPTER 3 Applications of Differentiation
Potential Energy and Conservation of Energy
Chapter 26 Current and Resistance
Extensions: Uncertainty, Risk Aversion, and Multiple Tasks
Introduction and Mathematical Concepts
EE 458 Introduction to Optimization
Chapter 1 – Essentials of Geometry
UNIT-3. Random Process – Temporal Characteristics
Performance Surfaces.
Chapter 4 Transients See notes on the chalkboard and the figures that follow.
Current and Resistance
Chapter 1 Functions.
1.6 Classify Polygons.
L7 Optimal Design concepts pt C
Presentation transcript:

Basic Concepts of Optimization Chapter 4 Basic Concepts of Optimization Chapter 4

Chapter 4 Objective Function issues: continuity of f (discrete cases, e.g., insulation, pipe sizes) (2) convexity, concavity stationary points (necessary condition) quadratic vs. non-quadratic functions scalar vs. vector case (6) sufficiency condition (min) Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4 FIGURE 4.9 Convex and nonconvex sets.

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4

Chapter 4