EE 458 Introduction to Optimization - PowerPoint Presentation

Slide1

EE 458Introduction to Optimization

J. McCalley

1Slide2

Real-time

Electricity markets and tools

Day-ahead

SCUC and SCED

SCED

Minimize f(

x

)

s

ubject to

h

(x)=cg(x)< b

BOTH LOOK LIKE THIS

SCUC: x contains discrete & continuous variables.

SCED:x contains only continuous variables.

2Slide3

Optimization Terminology

Minimize f(

x

)

s

ubject toh(x)=cg(x)< bf(x): Objective function x: Decision variablesh(x)=c: Equality constraintg(x)<

b: Inequality constraint

An optimization problem or a mathematical program or a mathematical programming problem.x*: solution3Slide4

Classification of Optimization Problems

http://www.neos-guide.org/NEOS/index.php/Optimization_Tree

Continuous Optimization

Unconstrained Optimization

Bound Constrained Optimization Derivative-Free Optimization Global Optimization Linear Programming Network Flow Problems Nondifferentiable Optimization Nonlinear Programming Optimization of Dynamic Systems Quadratic Constrained Quadratic Programming Quadratic Programming Second Order Cone Programming Semidefinite Programming Semiinfinite Programming Discrete and Integer Optimization

Combinatorial Optimization Traveling Salesman Problem

Integer Programming Mixed Integer Linear Programming Mixed Integer Nonlinear Programming Optimization Under Uncertainty Robust Optimization Stochastic Programming Simulation/Noisy Optimization Stochastic Algorithms Complementarity Constraints and Variational Inequalities Complementarity Constraints Game Theory Linear Complementarity Problems Mathematical Programs with Complementarity Constraints Nonlinear Complementarity Problems Systems of Equations Data Fitting/Robust Estimation Nonlinear Equations

Nonlinear Least Squares Systems of Inequalities Multiobjective Optimization 4Slide5

Convex functions

Definition #1

:

A

function f(x) is convex in an interval if its second derivative is positive on that interval.Example: f(x)=x2 is convex since f’(x)=2x, f’’(x)=2>05Slide6

Convex functions

The second derivative test is sufficient but not necessary.

www.ebyte.it/library/docs/math09/AConvexInequality.html

Definition #2

: A

function f(x) is convex if a line drawn between any two points on the function remains on or above the function in the interval between the two points.6Slide7

Convex functions

Definition #2

: A

function f(x) is convex if a line drawn between any two points on the function remains on or above the function in the interval between the two

points.

Is a linear function convex?Answer is “yes” since a line drawn between any two points on the function remains on the function.7Slide8

Convex Sets

Definition #3

:

A set C is convex if a line segment between any two points in C lies in C.

Ex

: Which of the below are convex sets?The set on the left is convex. The set on the right is not.8Slide9

Convex Sets

Definition #3

:

A set C is convex if a line segment between any two points in C lies in C.

S. Boyd and L.

Vandenberghe, “Convex optimization,” Cambridge University Press, 2004.9Slide10

Global vs. local optima

Example

: Solve the

following:

Minimize f(x)=x

2Solution: f’(x)=2x=0x*=0.This solution is a local optimum. It is also the global optimum.Example: Solve the following:Minimize f(x)=x3-17x2+80x-100Solution: f’(x)=3x2-34x+80=0Solving the above results in x=3.33 and x=8. Issue#1: Which is the best solution?Issue#2: Is the best solution the global solution?10Slide11

Global vs. local optima

Example

: Solve the

following:

Minimize f(x)=x

2Solution: f’(x)=2x=0x*=0.This solution is a local optimum. It is also the global optimum.Example: Solve the following:Minimize f(x)=x3-17x2+80x-100Solution: f’(x)=3x2-34x+80=0Solving the above results in x=3.33 and x=8. Issue#1: Which is the best solution?Issue#2: Is the best solution the global solution?11Slide12

Global vs. local optima

Example

: Solve the

following:

Minimize f(x)=x

3-17x2+80x-100Solution: f’(x)=3x2-34x+80=0. Solving results in x=3.33, x=8. Issue#1: Which is the best solution?Issue#2: Is the best solution the global solution?x=8

No! It is unbounded.12Slide13

Convexity & global vs. local optima



When

minimizing a function, if we want to be sure that we can get a global solution via differentiation, we need to impose some requirements on our objective function.



We will also need to impose some requirements on the set of possible values that the solution x* may take. Min f(x)subject toh(x)=cg(x)< b

Definition

: If f(x) is a convex function, and if S is a convex set, then the above problem is a convex programming problem. Definition: If f(x) is not a convex function, or if S is

not a convex set, then the above problem is a non-convex programming problem. 13Slide14

Convex vs. nonconvex

programming problems

The desirable quality of a convex programming problem is that any

locally optimal solution

is also a

globally optimal solution. If we have a method of finding a locally optimal solution, that method also finds for us the globally optimum solution.14The undesirable quality of a non-convex programming problem is that any method which finds a locally optimal solution does not necessarily find the globally optimum solution.MATHEMATICAL PROGRAMMINGConvex

Non-convex

We address convex programming problems for the remainder of these notes, and beyond.We will later address a special form of non-convex programming problems.Slide15

A convex programming problem

15

Two variables with one equality-constraint

Multi-variable with one equality-constraint.

Multi-variable with multiple equality-constraints.

We focus on this one, but conclusions we derive will also apply to the other two. The benefit of focusing on this one is that we can visualize it.Slide16

Contour maps

16

Definition

: A

contour map is a 2-dimensional plane, i.e., a coordinate system in 2

variables,

say, x1, x2, that illustrates curves (contours) of constant functional value f(x1, x2). Example

: Draw the contour map for .

[X,Y] = meshgrid

(-2.0:.2:2.0,-2.0:.2:2.0);Z = X.^2+Y.^2;[c,h]=contour(X,Y,Z);clabel(

c,h);grid;xlabel('x1');ylabel('x2');Slide17

Contour maps and 3-D illustrations

17

Example

: Draw the 3-D surface for

.

[X,Y] =

meshgrid

(-2.0:.2:2.0,-2.0:.2:2.0);

Z = X.^2+Y.^2;surfc(X,Y,Z)xlabel('x1')ylabel('x2')zlabel('f(x1,x2)')

Height is f(x)

Contours

Each

contour of fixed value

f

is the projection onto the

x1-x2

plane of a horizontal slice made of the 3-D figure at a value

f

above the

x1-x2

plane.Slide18

Solving a convex program: graphical analysis

18

Example

: Solve this

convex program:

.

Superimpose this relation on top of the contour plot for f(x

1

,x

2

).

1. f(x

1

,x

2

)

must be minimized, and so we would like the solution to be as close to the origin as possible;

2. The

solution must be on the thick line in the right-hand corner of the plot, since this line represents the equality constraint

.Slide19

Solving a convex program: graphical analysis

19

.

Solution:

Any

contour

f<3

does not intersect the equality constraint;

Any

contour

f>3

intersects the equality constraint at two points

.

The contour f=3 and the equality constraint just touch each other at the point

x

*.

“Just touch”:

The

two curves are

tangent

to one another at the solution point

. Slide20

Solving a convex program: graphical analysis

20

.

The

two curves are

tangent

to one another at the solution point

.



The normal (gradient) vectors of the two curves, at the solution (tangent) point, are parallel.

“Parallel” means that the two vectors have the same direction. We do not know that they have the same magnitude. To account for this, we equate with a “multiplier”

λ

:

This means the following two vectors are parallel:Slide21

Solving a convex program: graphical analysis

21

.

Moving everything to the left:

Alternately:

Performing the gradient operation (taking derivatives with respect to x

1

and x

2

) :

In this problem, we already know the solution, but what if we did not?

Then could we use the above equations to find the solution?Slide22

Solving a convex program: analytical analysis

22

In this problem, we already know the solution, but what if we did not?

Then could we use the above equations to find the solution?

NO! Because we only have 2 equations, yet 3 unknowns: x

1

, x

2

,

λ

.

So we need another equation. Where do we get that equation?

Recall our equality constraint:

h(x

1

, x

2

)-c=0

.

This must be satisfied! Therefore:

Three equations, three unknowns, we can solve.Slide23

Solving a convex program: analytical analysis

23

Observation: The three equations are simply partial derivatives of the function

This is obviously true for the first two equations

,

but it is not so obviously true for the last one. But to see it, observeSlide24

Formal approach to solving our problem

24

Define the

Lagrangian

function:

In a convex programming problem, the “first-order conditions” for finding the solution is given by

OR

Or more compactly

where we have used

x

=(x

1

, x

2

)Slide25

Formal approach to solving our problem

25

Define the

Lagrangian

function:

In a convex programming problem, the “first-order conditions” for finding the solution is given by

OR

Or more compactly

where we have used

x

=(x

1

, x

2

)