CE 191: Civil and Environmental Engineering Systems Analysis. LEC 05 : Optimality Conditions

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1 CE 191: Civil and Environmental Engineering Systems Analysis LEC : Optimality Conditions Professor Scott Moura Civil & Environmental Engineering University of California, Berkeley Fall 214 Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 1

2 Conditions for Optimality Consider an unconstrained QP min f(x) = 1 2 xt Qx + Rx Recall from calculus (e.g. Math 1A) the first order necessary condition (FONC) for optimality: If x is an optimum, then it must satisfy d dx f(x ) = = Qx + R = x = Q 1 R Also recall the second order sufficiency condition (SOSC): If x is a stationary point (i.e. it satisfies the FONC), then it is also a minimum if 2 f(x ) Q positive definite positive definite Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 2

3 Review: Positive-definite matrices All of the following conditions are equivalent: Consider Q R n n Q is positive definite x T Qx >, x the real parts of all eigenvalues of Q are positive Q is negative definite Q is positive semi-definite x T Qx, x the real parts of all eigenvalues of Q are positive, and at least one eigenvalue is zero Q is negative semi-definite Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 3

4 Nature of stationary point based on SOSC Hessian matrix Quadratic form Nature of x positive definite x T Qx > local minimum negative definite x T Qx < local maximum positive semi-definite x T Qx valley negative semi-definite x T Qx ridge indefinite x T Qx any sign saddle point Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 4

5 Local Minimum Local Minimum x 1 x 1 Q = [ , eig(q) = {1, } Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide

6 Local Maximum Local Maximum x x 1 Q = [ , eig(q) = { 3, 1} Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 6

7 Valley Valley x x 1 Q = [ , eig(q) = {, 1} Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 7

8 Ridge Ridge x x 1 Q = [, eig(q) = {, } Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 8

9 Saddle Point Saddle Point x x 2 x 1 Q = [ , eig(q) = { 2.26, 7.76} Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 9

10 Example.1 f(x 1, ) = (3 x 1 ) 2 + (4 ) 2 = [ [ [ 1 x1 x [ 6 8 [ x 1 Check the FONC: [ f 6 + x = 2x has the solution (x, 1 x 2 ) = (3, 4). = [ Check the SOFC: [ 2 f 2 x = 2 2 positive definite Solution: Unique local minimum Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 1

11 Example.2 f(x 1, ) = 4x x = [ [ [ 4 2 x1 x 1 + [ [ x Check the FONC: [ [ f 4 + x = 8x1 4 = 2 4x has an infinity of solutions (x 1, x 2 ) on the line 2x 1 = 1. Check the SOFC: 2 f = [ positive semidefinite Solution: Infinite set of minima (valley) Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 11

12 A Connection to Nonlinear Programming Consider the more general nonlinear programming problem, with nonlinear cost and nonlinear constraints min J = f(x) s. to g(x) Consider a given value for the decision variable, x k. Let s take a Taylor series expansion of the cost and constraints. Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 12

13 Taylor Series Review: Expand f(x) into infinite power series around x = x k f(x) = f(x k ) + f (x k ) (x x k ) f (x k ) (x x k ) = n= f n (x k ) n! (x x k ) n Expand cost function, truncated to be 2nd order f(x) f(x k ) + f (x k ) (x x k ) f (x k ) (x x k ) 2 Expand inequality constraints, truncated to be 1st order g(x) g(x k ) + g (x k ) (x x k ) For ease of notation, define x = x x k Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 13

14 Sequential Quadratic Programming (SQP) We arrive at the following approximate QP min Q + R x s. to A x b where Q = 1 2 f (x k ), R = f (x k ) A = g (x k ), b = g(x k ) Suppose the optimal solution is x. Then let x k+1 = x k + x. Repeat. Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 14

15 SQP Remarks Remark 1: Can add equality constraints h(x) = and expand via 1st order Taylor series. Remark 2: If x k+1 does not satisfy g(x k+1 ), then you can project x k+1 onto surface g( ) =. Remark 3: Iterate until a stopping criteria is reached, e.g. x ε. Summary: Can re-formulate nonlinear program into sequence of quadratic programs. Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 1

16 Example.3 Consider the NLP min e x1 + ( 2) 2 x 1, s. to x 1 1. with the initial guess [x 1,,, T = [1, 1 T. Perform 3 iterations of SQP. We have the functions: f(x) = e x1 + ( 2) 2 and g(x) = x 1 1 We seek to find the approximate QP subproblem min 1 2 xt Q x + R T x s. to A x b Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 16

17 Example.3 Taking derivatives of f(x) and g(x), Q = [ e x 1 2 [, R = A = [ x 1, b = 1 x1 e x1 2( 2) Now consider the initial guess [x 1,,, T = [1, 1 T. This iterate is feasible., First iteration: Q, R, A, b matrices are [ [ e 1 e 1 Q =, R = 2 2 A = [ 1 1, b =, Solving this QP subproblem results in x = [.6893, Next iterate: [x 1,1,,1 = [x 1,,, + x = [.317, Iterate is feasible. Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 17

18 Example.3 Second iteration: Result is [x 1,2,,2 = [.443, Iterate is infeasible. Continuing the process g(x) = Iterations of SQP x Iter. [x 1, f(x) g(x) [1, [.317, [.443, [.22, [.211, Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 18

19 Additional Reading Papalambros & Wilde Section Local Approximations Papalambros & Wilde Section Optimality Conditions Papalambros & Wilde Section Sequential Quadratic Programming Prof. Moura UC Berkeley CE 191 LEC 4 - Optimality Conditions Slide 19

CHAPTER 2: QUADRATIC PROGRAMMING

CHAPTER 2: QUADRATIC PROGRAMMING Overview Quadratic programming (QP) problems are characterized by objective functions that are quadratic in the design variables, and linear constraints. In this sense,