Lecture 9 – Nonlinear Programming Models Topics Convex sets and convex programming First-order optimality conditions Examples Problem classes

General NLP Minimize f(x) s.t. gi(x) (, , =) bi, i = 1,…,m x = (x1,…,xn)T is the n-dimensional vector of decision variables f (x) is the objective function gi(x) are the constraint functions bi are fixed known constants

x0 = lx1 + (1–l)x2 Î S for all l such 0 ≤ l ≤ 1. Convex Sets Definition: A set S  n is convex if every point on the line segment connecting any two points x1, x2 Î S is also in S. Mathematically, this is equivalent to x0 = lx1 + (1–l)x2 Î S for all l such 0 ≤ l ≤ 1.  x1 x2 x1 x1   x2   x2 

(Nonconvex) Feasible Region S = {(x1, x2) : (0.5x1 – 0.6)x2 ≤ 1 2(x1)2 + 3(x2)2 ≥ 27; x1, x2 ≥ 0}

Convex Sets and Optimization Let S = { x Î n : gi(x) £ bi, i = 1,…,m } Fact: If gi(x) is a convex function for each i = 1,…,m then S is a convex set. Convex Programming Theorem: Let x  n and let f (x) be a convex function defined over a convex constraint set S. If a finite solution exists to the problem Minimize { f (x) : x Î S } then all local optima are global optima. If f (x) is strictly convex, the optimum is unique.

Convex Programming Min f (x1,…,xn) s.t. gi(x1,…,xn) £ bi i = 1,…,m is a convex program if f is convex and each gi is convex. Max f (x1,…,xn) s.t. gi(x1,…,xn) £ bi i = 1,…,m x1 ³ 0,…,xn ³ 0 is a convex program if f is concave and each gi is convex.

Linearly Constrained Convex Function with Unique Global Maximum Maximize f (x) = (x1 – 2)2 + (x2 – 2)2 subject to –3x1 – 2x2 ≤ –6 –x1 + x2 ≤ 3 x1 + x2 ≤ 7 2x1 – 3x2 ≤ 4

(Nonconvex) Optimization Problem

First-Order Optimality Conditions Minimize { f (x) : gi(x)  bi, i = 1,…,m } Lagrangian: Optimality conditions Stationarity: Complementarity: migi(x) = 0, i = 1,…,m Feasibility: gi(x)  bi, i = 1,…,m Nonnegativity: mi  0, i = 1,…,m

Importance of Convex Programs Commercial optimization software cannot guarantee that a solution is globally optimal to a nonconvex program. NLP algorithms try to find a point where the gradient of the Lagrangian function is zero – a stationary point – and complementary slackness holds. Given L(x,m) = f(x) + m(g(x) – b) we want L(x,m) = f(x) + mg(x) = 0 m(g(x) – b) = 0 g(x) – b ≤ 0, m ³ 0 For a convex program, all local solutions are global optima.

Example: Cylinder Design We want to build a cylinder (with a top and a bottom) of maximum volume such that its surface area is no more than s units. Max V(r,h) = pr2h s.t. 2pr2 + 2prh = s r ³ 0, h ³ 0 r h There are a number of ways to approach this problem. One way is to solve the surface area constraint for h and substitute the result into the objective function.

Solution by Substitution s - 2pr 2 s - 2pr 2 rs  Volume = V = pr2 - pr 3 h = [ ] = 2r 2pr 2 dV s s s 1/2 1/2 = 0  r = ( ) h = - r = 2( ) dr 6 p 2pr 6 p s 3/2 s 1/2 s 1/2 V = pr 2h = 2p ( ) p r = ( ) h = 2( ) 6 6 p 6 Is this a global optimal solution?

Test for Convexity rs dV(r) s d2V(r ) - pr 3  = - 3pr 2  = 6pr dr 2 dr 2 d 2 V £ 0 for all r ³ 0 dr 2 Thus V(r ) is concave on r ³ 0 so the solution is a global maximum.

Advertising (with Diminishing Returns) A company wants to advertise in two regions. The marketing department says that if $x1 is spent in region 1, sales volume will be 6(x1)1/2. If $x2 is spent in region 2, sales volume will be 4(x2)1/2. The advertising budget is $100. Model: Max f (x) = 6(x1)1/2 + 4(x2)1/2 s.t. x1 + x2 £ 100, x1 ³ 0, x2 ³ 0 Solution: x1* = 69.2, x2* = 30.8, f (x*) = 72.1 Is this a global optimum?

Excel Add-in Solution

Portfolio Selection with Risky Assets (Markowitz) Suppose that we may invest in (up to) n stocks. Investors worry about (1) expected gain (2) risk. Let rj = random variable associated with return on stock j mj = expected return on stock j sjj = variance of return for stock j We are also concerned with the covariance terms: sij = cov(ri, rj) If sij > 0 then returns on i and j are positively correlated. If sij < 0 returns are negatively correlated.

Decision Variables: xj = # of shares of stock j purchased Expected return of the portfolio: R(x) = å mjxj n i =1 n j =1 Variance (measure of risk): V(x) = å  å sijxixj V(x) = s11x1x1 + s12x1x2 + s21x2x1 + s22x2x1 = 1 + (-1) + (-1) + 1 = 0 Thus we can construct a “risk-free” portfolio (from variance point of view) if we can find stocks “fully” negatively correlated. Example: If x1 = x2 = 1, we get

Nonlinear optimization models … If , then buying stock 2 is just like buying additional shares of stock 1. Nonlinear optimization models … Let pj = price of stock j b = our total budget b = risk-aversion factor (when b = 0 risk is not a factor) Consider 3 different models: 1) Max f (x) = R(x) – bV(x) s.t. å pj xj £ b, xj ³ 0, j = 1,…,n where b ³ 0 is determined by the decision maker n j =1

s.t. V(x) £ a, å pjxj £ b, xj ³ 0, j = 1,…,n Max f (x) = R(x) s.t. V(x) £ a, å pjxj £ b, xj ³ 0, j = 1,…,n where a ³ 0 is determined by the investor. Smaller values of a represent greater risk aversion. n j =1 3) Min f (x) = V(x) s.t. R(x) ³ g, å pj xj £ b, xj ³ 0, j = 1,…,n where g ³ 0 is the desired rate of return (minimum expectation) is selected by the investor. n j =1

Hanging Chain with Rigid Links 10ft x y 1 ft each link What is equilibrium shape of chain? Decision variables: Let (xj, yj), j = 1,…,n, be the incremental horizontal and vertical displacement of each link, where n ³ 10. Constraints: xj2 + yj2 = 1, j = 1,…,n, each link has length 1 x1 + x2 + • • • + xn = 10, net horizontal displacement y1 + y2 + • • • + yn = 0, net vertical displacement

Summary Objective: Minimize chain’s potential energy Assuming that the center of the mass of each link is at the center of the link. This is equivalent to minimizing 1 2 y1 + (y1 + y2) + (y1 + y2 + y3) + • • • (y1 + • • • + yn-1 + yn) = (n - 1 + 1 2 ]y1 + (n - 2 + )y2 + (n - 3 + )y3 + • • • + 3 yn-1 + yn Summary Min å (n - j + ½)yj n j =1 s.t. xj2 + yj2 = 1, j = 1,…,n x1 + x2 + • • • + xn = 10 y1 + y2 + • • • + yn = 0

Is a local optimum guaranteed to be a global optimum? No! Consider a chain with 4 links: These solutions are both local minima. Constraints xj2 + yj2 = 1 for all j yield a nonconvex feasible region so there may be several local optima.

Direct Current Network Problem: Determine the current flows I1, I2,…,I7 so that the total content is minimized Content: G(I) = 0I v(i)di for I ≥ 0 and G(I) = I 0 v(i)di for I < 0

Solution Approach Electrical Engineering: Use Kirchoff’s laws to find currents when power source is given. Operations Research: Optimize performance measure in network taking flow balance into account. Linear resistor: Voltage, v(I ) = IR Content function, G(I ) = I 2R/2 Battery: Voltage, v(I ) = –E Content function, G(I ) = –EI

Network Flow Model Network diagram: Minimize Z = –100I1 + 5I22 + 5I32 + 10I42 + 10I52 subject to I1 – I2 = 0, I2 – I3 – I4 = 0, I5 – I6 = 0, I5 + I7 = 0, I3 + I6 – I7 = 0, –I1 – I6 = 0 Solution: I1 = I2 = 50/9, I3 = 40/9, I4 = I5 = 10/9, I6 = –50/9, I7 = –10/9

NLP Problem Classes Constrained vs. unconstrained Convex programming problem Quadratic programming problem f (x) = a + cTx + ½ xTQx, Q  0 Separable programming problem f (x) = j=1,n fj(xj) Geometric programming problem g(x) = t=1,T ctPt(x), Pt(x) = (x1at1) . . . (xnatn), xj > 0 Equality constrained problems

What You Should Know About Nonlinear Programming How to identify a convex program. How to write out the first-order optimality conditions. The difference between a local and global solution. How to classify problems.