Lecture 1 Introduction and overview

Transcription

1 ESE504 (Fall 2010) Lecture 1 Introduction and overview linear programming example course topics software integer linear programming 1 1

2 Linear program (LP) minimize subject to n c j x j j=1 n a ij x j b i, j=1 n c ij x j = d i, j=1 i =1,...,m i =1,...,p variables: x j problem data: the coefficients c j, a ij, b i, c ij, d i can be solved very efficiently (several 10,000 variables, constraints) widely available general-purpose software extensive, useful theory (optimality conditions, sensitivity analysis,... ) Introduction and overview 1 2

3 Example. Open-loop control problem single-input/single-output system (with input u, output y) y(t) =h 0 u(t)+h 1 u(t 1) + h 2 u(t 2) + h 3 u(t 3) + output tracking problem: minimize deviation from desired output y des (t) max t=0,...,n y(t) y des(t) subject to input amplitude and slew rate constraints: u(t) U, u(t +1) u(t) S variables: u(0),..., u(m) (with u(t) =0for t < 0, t > M) solution: can be formulated as an LP, hence easily solved (more later) Introduction and overview 1 3

4 example step response (s(t) =h t + + h 0 ) and desired output: step response y des (t) amplitude and slew rate constraint on u: u(t) 1.1, u(t) u(t 1) 0.25 Introduction and overview 1 4

5 optimal solution 1 output and desired output 1.1 input u(t) u(t) u(t 1) Introduction and overview 1 5

6 Brief history 1930s (Kantorovich): economic applications 1940s (Dantzig): military logistics problems during WW2; 1947: simplexalgorithm 1950s 60s discovery of applications in many other fields (structural optimization, control theory, filter design,... ) 1979 (Khachiyan) ellipsoid algorithm: more efficient (polynomial-time) than simplex in worst case, but slower in practice 1984 (Karmarkar): projective (interior-point) algorithm: polynomial-time worst-case complexity, and efficient in practice 1984 today. many variations of interior-point methods (improved complexity or efficiency in practice), software for large-scale problems Introduction and overview 1 6

7 Course outline the linear programming problem linear inequalities, geometry of linear programming engineering applications signal processing, control, structural optimization... duality algorithms the simplex algorithm, interior-point algorithms large-scale linear programming and network optimization techniques for LPs with special structure, network flow problems integer linear programming introduction, some basic techniques Introduction and overview 1 7

8 Software solvers: solve LPs described in some standard form modeling tools: accept a problem in a simpler, more intuitive, notation and convert it to the standard form required by solvers software for this course (see class website) platforms: Matlab, Octave, Python solvers: linprog (Matlab Optimization Toolbox), modeling tools: CVX (Matlab), YALMIP (Matlab), Thanks to Lieven Vandenberghe at UCLA for his slides Introduction and overview 1 8

9 integer linear program Integer linear program minimize subject to n j=1 c jx j n j=1 a ijx j b i, i =1,...,m n j=1 c ijx j = d i, i =1,...,p x j Z Boolean linear program minimize subject to n j=1 c jx j n j=1 a ijx j b i, i =1,...,m n j=1 c ijx j = d i, i =1,...,p x j {0, 1} very general problems; can be extremely hard to solve can be solved as a sequence of linear programs Introduction and overview 1 9

10 scheduling graph V: Example. Scheduling problem i j n nodes represent operations (e.g., jobs in a manufacturing process, arithmetic operations in an algorithm) (i, j) V means operation j must wait for operation i to be finished M identical machines/processors; each operation takes unit time problem: determine fastest schedule Introduction and overview 1 10

11 Boolean linear program formulation variables: x is, i =1,...,n, s =0,...,T: x is =1if job i starts at time s, x is =0otherwise constraints: 1. x is {0, 1} 2. job i starts exactly once: T x is =1 s=0 3. if there is an arc (i, j) in V, then T sx js s=0 T sx is 1 s=0 Introduction and overview 1 11

12 4. limit on capacity (M machines) at time s: n x is M i=1 cost function (start time of job n): T s=0 sx ns Boolean linear program minimize subject to T s=0 sx ns T s=0 x is =1, i =1,...,n T s=0 sx js T s=0 sx is 1, (i, j) V n i=1 x is M, s =0,...,T x is {0, 1}, i =1,...,n, s=0,...,t Introduction and overview 1 12

13 ESE504 (Fall 2010) Lecture 2 Linear inequalities vectors inner products and norms linear equalities and hyperplanes linear inequalities and halfspaces polyhedra 2 1

14 Vectors (column) vector x R n : x = x 1 x 2. x n x i R: ith component or element of x also written as x =(x 1,x 2,...,x n ) some special vectors: x =0(zero vector): x i =0, i =1,...,n x = 1: x i =1, i =1,...,n x = e i (ith basis vector or ith unit vector): x i =1, x k =0for k i (n follows from context) Linear inequalities 2 2

15 Vector operations multiplying a vector x R n with a scalar α R: αx = αx 1. αx n adding and subtracting two vectors x, y R n : x + y = x 1 + y 1. x n + y n, x y = x 1 y 1. x n y n y 1.5y 0.75x +1.5y 0.75x x Linear inequalities 2 3

16 Inner product x, y R n x, y := x 1 y 1 + x 2 y x n y n = x T y important properties αx, y = α x, y x + y, z = x, z + y, z x, y = y, x x, x 0 x, x =0 x =0 linear function: f : R n R is linear, i.e. f(αx + βy) =αf(x)+βf(y), if and only if f(x) = a, x for some a Linear inequalities 2 4

17 Euclidean norm for x R n we define the (Euclidean) norm as x = x x x2 n = x T x x measures length of vector (from origin) important properties: αx = α x (homogeneity) x + y x + y (triangle inequality) x 0 (nonnegativity) x =0 x =0(definiteness) distance between vectors: dist(x, y) = x y Linear inequalities 2 5

18 angle between vectors in R n : i.e., x T y = x y cos θ Inner products and angles θ = (x, y) =cos 1 x and y aligned: θ =0; x T y = x y x and y opposed: θ = π; x T y = x y x T y x y x and y orthogonal: θ = π/2 or π/2; x T y =0(denoted x y) x T y>0 means (x, y) is acute; x T y<0 means (x, y) is obtuse x x x T y>0 y x T y<0 y Linear inequalities 2 6

19 Cauchy-Schwarz inequality: x T y x y projection of x on y x y θ ( x T ) y y 2 y projection is given by ( x T ) y y 2 y Linear inequalities 2 7

20 Hyperplanes hyperplane in R n : {x a T x = b} (a 0) solution set of one linear equation a 1 x a n x n = b with at least one a i 0 set of vectors that make a constant inner product with vector a =(a 1,...,a n ) (the normal vector) x 0 a ( at x 0 a 2 )a 0 x (a T x = a T x 0 ) in R 2 : a line, in R 3 :aplane,... Linear inequalities 2 8

21 Halfspaces (closed) halfspace in R n : {x a T x b} (a 0) solution set of one linear inequality a 1 x a n x n b with at least one a i 0 a =(a 1,...,a n ) is the (outward) normal x 0 a 0 {x a T x a T x 0 } {x a T x a T x 0 } {x a T x<b} is called an open halfspace Linear inequalities 2 9

22 Affine sets solution set of a set of linear equations a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 1 a m1 x 1 + a m2 x a mn x n = b m intersection of m hyperplanes with normal vectors a i =(a i1,a i2,...,a in ) (w.l.o.g., all a i 0). in matrix notation: with A = Ax = b a 11 a 12 a 1n a 21. a 22. a 2n. a m1 a m2 a mn, b = b 1 b 2. b m Linear inequalities 2 10

23 Polyhedra solution set of system of linear inequalities a 11 x 1 + a 12 x a 1n x n b 1 a m1 x 1 + a m2 x a mn x n b m intersection of m halfspaces, with normal vectors a i =(a i1,a i2,...,a in ) (w.l.o.g., all a i 0) a 1 a 2. a 5 a 3 a 4 Linear inequalities 2 11

24 matrix notation Ax b with A = a 11 a 12 a 1n a 21. a 22. a 2n. a m1 a m2 a mn, b = b 1 b 2. b m Ax b stands for componentwise inequality, i.e., fory, z R n, y z y 1 z 1,...,y n z n Linear inequalities 2 12

25 Examples of polyhedra ahyperplane{x a T x = b}: a T x b, a T x b solution set of system of linear equations/inequalities a T i x b i, i =1,...,m, c T i x = d i, i =1,...,p aslab{x b 1 a T x b 2 } the probability simplex {x R n 1 T x =1, x i 0, i=1,...,n} (hyper)rectangle {x R n l x u} where l<u Linear inequalities 2 13

26 Linear inequalities 2 14

27 ESE504 (Fall 2010) Lecture 3 Geometry of linear programming subspaces and affine sets, independent vectors matrices, range and nullspace, rank, inverse polyhedron in inequality form extreme points degeneracy the optimal set of a linear program 3 1

28 Subspaces S R n (S ) is called a subspace if x, y S, α,β R = αx + βy S αx + βy is called a linear combination of x and y examples (in R n ) S= R n, S = {0} S={αv α R} where v R n (i.e., a line through the origin) S= span(v 1,v 2,...,v k )={α 1 v α k v k α i R}, wherev i R n set of vectors orthogonal to given vectors v 1,...,v k : S = {x R n v1 T x =0,...,vk T x =0} Geometry of linear programming 3 2

29 Independent vectors vectors v 1,v 2,...,v k are independent if and only if α 1 v 1 + α 2 v α k v k =0 = α 1 = α 2 = =0 some equivalent conditions: coefficients of α 1 v 1 + α 2 v α k v k are uniquely determined, i.e., α 1 v 1 + α 2 v α k v k = β 1 v 1 + β 2 v β k v k implies α 1 = β 1,α 2 = β 2,...,α k = β k no vector v i can be expressed as a linear combination of the other vectors v 1,...,v i 1,v i+1,...,v k Geometry of linear programming 3 3

30 Basis and dimension {v 1,v 2,...,v k } is a basis for a subspace S if v 1,v 2,...,v k span S, i.e., S = span(v 1,v 2,...,v k ) v 1,v 2,...,v k are independent equivalently: every v Scan be uniquely expressed as v = α 1 v α k v k fact: for a given subspace S, the number of vectors in any basis is the same, and is called the dimension of S, denoted dim S Geometry of linear programming 3 4

31 Affine sets V R n (V ) is called an affine set if x, y V,α+ β =1 = αx + βy V αx + βy is called an affine combination of x and y examples (in R n ) subspaces V= b + S = {x + b x S}where S is a subspace V= {α 1 v α k v k α i R, i α i =1} V= {x v T 1 x = b 1,...,v T k x = b k} (if V ) every affine set V canbewrittenasv = x 0 + S where x 0 R n, S a subspace (e.g., can take any x 0 V, S = V x 0 ) dim(v x 0 ) is called the dimension of V Geometry of linear programming 3 5

32 A = some special matrices: Matrices a 11 a 12 a 1n a 21. a 22. a 2n. a m1 a m2 a mn Rm n A =0(zero matrix): a ij =0 A = I (identity matrix): m = n and A ii =1for i =1,...,n, A ij =0 for i j A = diag(x) where x R n (diagonal matrix): m = n and A = x x x n Geometry of linear programming 3 6

33 Matrix operations addition, subtraction, scalar multiplication transpose: A T = a 11 a 21 a m1 a 12. a 22. a m2. a 1n a 2n a mn Rn m multiplication: A R m n, B R n q, AB R m q : AB = n i=1 a n 1ib i1 i=1 a n 1ib i2 i=1 a n 1ib iq i=1 a n 2ib i1 i=1 a 2ib i2 n i=1 a 2ib iq... n i=1 a n mib i1 i=1 a n mib i2 i=1 a mib iq Geometry of linear programming 3 7

34 Rows and columns rows of A R m n : A = a T 1 a T 2. with a i =(a i1,a i2,...,a in ) R n a T m columns of B R n q : B = [ b 1 b 2 b q ] with b i =(b 1i,b 2i,...,b ni ) R n for example, can write AB as AB = a T 1 b 1 a T 1 b 2 a T 1 b q a T 2 b 1. a T 2 b 2. a T 2 b q. a T mb 1 a T mb 2 a T mb q Geometry of linear programming 3 8

35 Range of a matrix the range of A R m n is defined as R(A) ={Ax x R n } R m a subspace set of vectors that can be hit by mapping y = Ax the span of the columns of A =[a 1 a n ] R(A) ={a 1 x a n x n x R n } the set of vectors y s.t. Ax = y has a solution R(A) =R m Ax = y can be solved in x for any y the columns of A span R m dim R(A) =m Geometry of linear programming 3 9

36 Interpretations v R(A), w R(A) y = Ax represents output resulting from input x v is a possible result or output w cannot be a result or output R(A) characterizes the achievable outputs y = Ax represents measurement of x y = v is a possible or consistent sensor signal y = w is impossible or inconsistent; sensors have failed or model is wrong R(A) characterizes the possible results Geometry of linear programming 3 10

37 Nullspace of a matrix the nullspace of A R m n is defined as N (A) ={ x R n Ax =0} a subspace the set of vectors mapped to zero by y = Ax the set of vectors orthogonal to all rows of A: where A =[a 1 a m ] T zero nullspace: N (A) ={0} N (A) = { x R n a T 1 x = = a T mx =0 } x can always be uniquely determined from y = Ax (i.e., the linear transformation y = Ax doesn t lose information) columns of A are independent Geometry of linear programming 3 11

38 Interpretations suppose z N(A) y = Ax represents output resulting from input x z is input with no result x and x + z have same result N (A) characterizes freedom of input choice for given result y = Ax represents measurement of x z is undetectable get zero sensor readings x and x + z are indistinguishable: Ax = A(x + z) N (A) characterizes ambiguity in x from y = Ax Geometry of linear programming 3 12

39 Inverse A R n n is invertible or nonsingular if det A 0 equivalent conditions: columns of A are a basis for R n rows of A are a basis for R n N(A) ={0} R(A) =R n y = Ax has a unique solution x for every y R n A has an inverse A 1 R n n,withaa 1 = A 1 A = I Geometry of linear programming 3 13

40 Rank of a matrix we define the rank of A R m n as rank(a) =dimr(a) (nontrivial) facts: rank(a) =rank(a T ) rank(a) is maximum number of independent columns (or rows) of A, hence rank(a) min{m, n} rank(a)+dimn (A) =n Geometry of linear programming 3 14

41 Full rank matrices for A R m n we have rank(a) min{m, n} we say A is full rank if rank(a) =min{m, n} for square matrices, full rank means nonsingular for skinny matrices (m >n), full rank means columns are independent for fat matrices (m <n), full rank means rows are independent Geometry of linear programming 3 15

42 Sets of linear equations given A R m n, y R m Ax = y solvable if and only if y R(A) unique solution if y R(A) and rank(a) =n general solution set: {x 0 + v v N(A)} where Ax 0 = y A square and invertible: unique solution for every y: x = A 1 y Geometry of linear programming 3 16

43 A =[a 1 a m ] T R m n, b R m Polyhedron (inequality form) P = {x Ax b} = {x a T i x b i, i =1,...,m} a 1 a 2 a 6 a 5 a 3 P is convex: a 4 x, y P, 0 λ 1 = λx +(1 λ)y P i.e., theline segment between any two points in P lies in P Geometry of linear programming 3 17

44 Extreme points and vertices x P is an extreme point if it cannot be written as x = λy +(1 λ)z with 0 λ 1, y, z P, y x, z x P c c T x constant x P is a vertex if there is a c such that c T x<c T y for all y P, y x fact: x is an extreme point x is a vertex (proof later) Geometry of linear programming 3 18

45 Basic feasible solution define I as the set of indices of the active or binding constraints (at x ): a T i x = b i, i I, a T i x <b i, i I define Ā as Ā = a T i 1 a T i 2. a T i k, I = {i 1,...,i k } x is called a basic feasible solution if rank A = n fact: x is a vertex (extreme point) x is a basic feasible solution (proof later) Geometry of linear programming 3 19

46 Example x (1,1) is an extreme point (1,1) is a vertex: unique minimum of c T x with c =( 1, 1) (1,1) is a basic feasible solution: I = {2, 4} and rank A =2,where A = [ ] Geometry of linear programming 3 20

47 Equivalence of the three definitions vertex = extreme point let x be a vertex of P, i.e., thereisac 0such that c T x <c T x for all x P, x x let y, z P, y x, z x : c T x <c T y, c T x <c T z so, if 0 λ 1, then hence x λy +(1 λ)z c T x <c T (λy +(1 λ)z) Geometry of linear programming 3 21

48 extreme point = basic feasible solution suppose x P is an extreme point with a T i x = b i, i I, a T i x <b i, i I suppose x is not a basic feasible solution; then there exists a d 0with a T i d =0, i I and for small enough ɛ>0, y = x + ɛd P, z = x ɛd P we have x =0.5y +0.5z, which contradicts the assumption that x is an extreme point Geometry of linear programming 3 22

49 basic feasible solution = vertex suppose x P is a basic feasible solution and a T i x = b i i I, a T i x <b i i I define c = i I a i;then c T x = i I b i and for all x P, c T x i I b i with equality only if a T i x = b i, i I however the only solution to a T i x = b i, i I, isx ; hence c T x <c T x for all x P Geometry of linear programming 3 23

50 Degeneracy set of linear inequalities a T i x b i, i =1,...,m a basic feasible solution x with a T i x = b i, i I, a T i x <b i, i I is degenerate if #indices in I is greater than n a property of the description of the polyhedron, not its geometry affects the performance of some algorithms disappears with small perturbations of b Geometry of linear programming 3 24

51 Unbounded directions P contains a half-line if there exists d 0, x 0 such that x 0 + td P for all t 0 equivalent condition for P = {x Ax b}: Ax 0 b, Ad 0 fact: P unbounded P contains a half-line P contains a line if there exists d 0, x 0 such that x 0 + td P for all t equivalent condition for P = {x Ax b}: Ax 0 b, Ad =0 fact: P has no extreme points P contains a line Geometry of linear programming 3 25

52 Optimal set of an LP minimize subject to c T x Ax b optimal value: p =min{c T x Ax b} (p = ± is possible) optimal point: x with Ax b and c T x = p optimal set: X opt = {x Ax b, c T x = p } example minimize c 1 x 1 + c 2 x 2 subject to 2x 1 + x 2 1 x 1 0, x 2 0 c =(1, 1): X opt = {(0, 0)}, p =0 c =(1, 0): X opt = {(0,x 2 ) 0 x 2 1}, p =0 c =( 1, 1): X opt =, p = Geometry of linear programming 3 26

53 Existence of optimal points p = if and only if there exists a feasible half-line with c T d<0 {x 0 + td t 0} d x 0 p =+ if and only if P = p is finite if and only if X opt c Geometry of linear programming 3 27

54 property: if P has at least one extreme point and p is finite, then there exists an extreme point that is optimal c X opt Geometry of linear programming 3 28

55 ESE504 (Fall 2010) Lecture 4 The linear programming problem: variants and examples variants of the linear programming problem LP feasibility problem examples and some general applications linear-fractional programming 4 1

56 Variants of the linear programming problem general form minimize c T x subject to a T i x b i, i =1,...,m gi T x = h i, i =1,...,p in matrix notation: where A = a T 1 a T 2. minimize subject to c T x Ax b Gx = h Rm n, G = g T 1 g T 2. Rp n a T m g T p The linear programming problem: variants and examples 4 2

57 inequality form LP minimize c T x subject to a T i x b i, i =1,...,m in matrix notation: minimize subject to c T x Ax b standard form LP minimize c T x subject to gi T x = h i, i =1,...,m x 0 in matrix notation: minimize subject to c T x Gx = h x 0 The linear programming problem: variants and examples 4 3

58 Reduction of general LP to inequality/standard form minimize c T x subject to a T i x b i, i =1,...,m gi T x = h i, i =1,...,p reduction to inequality form: minimize c T x subject to a T i x b i, i =1,...,m gi T x h i, i =1,...,p gi T x h i, i =1,...,p in matrix notation (where A has rows a T i, G has rows gt i ) minimize subject to c T x A G G x b h h The linear programming problem: variants and examples 4 4

59 reduction to standard form: minimize c T x + c T x subject to a T i x+ a T i x + s i = b i, i =1,...,m gi T x+ gi T x = h i, i =1,...,p x +,x,s 0 variables x +, x, s recover x as x = x + x s R m is called a slack variable in matrix notation: where x = x+ x s, c = minimize subject to c c 0, G = c T x G x = h x 0 [ A A I G G 0 ] [ b, h = h ] The linear programming problem: variants and examples 4 5

60 LP feasibility problem feasibility problem: find x that satisfies a T i x b i, i =1,...,m solution via LP (with variables t, x) minimize t subject to a T i x b i + t, i =1,...,m variables t, x if minimizer x, t satisfies t 0, thenx satisfies the inequalities LP in matrix notation: x = [ x t minimize c T x subject to à x b ] [ ] 0, c =, 1 à = [ A 1 ], b = b The linear programming problem: variants and examples 4 6

61 Piecewise-linear minimization piecewise-linear minimization: minimize max i=1,...,m (c T i x + d i) max i (c T i x + d i) c T i x + d i x equivalent LP (with variables x R n, t R): minimize t subject to c T i x + d i t, i =1,...,m in matrix notation: x = [ x t ], c = minimize c T x subject to à x [ ] b 0, à = [ C 1 ] [ ], b = d 1 The linear programming problem: variants and examples 4 7

62 Convex functions f : R n R is convex if for 0 λ 1 f(λx +(1 λ)y) λf(x)+(1 λ)f(y) λf(x) +(1 λ)f(y) x λx +(1 λ)y y The linear programming problem: variants and examples 4 8

63 Piecewise-linear approximation assume f : R n R differentiable and convex 1st-order approximation at x 1 is a global lower bound on f: f(x) f(x 1 )+ f(x 1 ) T (x x 1 ) f(x) x 1 x evaluating f, f at several x i yields a piecewise-linear lower bound: f(x) ( max f(x i )+ f(x i ) T (x x i ) ) i=1,...,k The linear programming problem: variants and examples 4 9

64 Convex optimization problem minimize (f i convex and differentiable) f 0 (x) LP approximation (choose points x j, j =1,...,K): minimize t subject to f 0 (x j )+ f 0 (x j ) T (x x j ) t, j =1,...,K (variables x, t) yields lower bound on optimal value can be extended to nondifferentiable convex functions more sophisticated variation: cutting-plane algorithm (solves convex optimization problem via sequence of LP approximations) The linear programming problem: variants and examples 4 10

65 Norms norms on R n : Euclidean norm x (or x 2 ) = x x2 n l 1 -norm: x 1 = x x n l - (or Chebyshev-) norm: x =max i x i x x =1 x =1 x 1 =1 x 1 The linear programming problem: variants and examples 4 11

66 Norm approximation problems minimize Ax b p x R n is variable; A R m n and b R m are problem data p =1, 2, r = Ax b is called residual r i = a T i x b i is ith residual (a T i is ith row of A) usually overdetermined, i.e., b R(A) (e.g., m>n, A full rank) interpretations: approximate or fit b with linear combination of columns of A b is corrupted measurement of Ax; find least inconsistent value of x for given measurements The linear programming problem: variants and examples 4 12

67 examples: r = r T r: least-squares or l 2 -approximation (a.k.a. regression) r =max i r i : Chebyshev, l, or minimax approximation r = i r i : absolute-sum or l 1 -approximation solution: l 2 : closed form expression x opt =(A T A) 1 A T b (assume rank(a) =n) l 1, l : no closed form expression, but readily solved via LP The linear programming problem: variants and examples 4 13

68 l 1 -approximation problem l 1 -approximation via LP minimize Ax b 1 write as minimize subject to m i=1 y i y Ax b y an LP with variables y, x: minimize subject to c T x à x b with x = [ x y ], c = [ 0 1 ], à = [ A A I I ] [, b = b b ] The linear programming problem: variants and examples 4 14

69 l -approximation problem l -approximation via LP minimize Ax b write as minimize subject to t t1 Ax b t1 an LP with variables t, x: minimize subject to c T x à x b with x = [ x t ], c = [ 0 1 ], à = [ A A 1 1 ] [, b = b b ] The linear programming problem: variants and examples 4 15

70 Example minimize Ax b p for p =1, 2, (A R ) resulting residuals: ri (p =1) ri (p =2) ri (p = ) i i i The linear programming problem: variants and examples 4 16

71 histogram of residuals: number of ri number of ri number of ri r (p =1) r (p =2) r (p = ) p = gives thinnest distribution; p = 1gives widest distribution p =1most very small (or even zero) r i The linear programming problem: variants and examples 4 17

72 Interpretation: maximum likelihood estimation m linear measurements y 1,...,y m of x R n : y i = a T i x + v i, i =1,...,m v i : measurement noise, IID with density p y is a random variable with density p x (y) = m i=1 p(y i a T i x) log-likelihood function is defined as log p x (y) = m log p(y i a T i x) i=1 maximum likelihood (ML) estimate of x is ˆx =argmax x m log p(y i a T i x) i=1 The linear programming problem: variants and examples 4 18

73 examples v i Gaussian: p(z) =1/( 2πσ)e z2 /2σ 2 ML estimate is l 2 -estimate ˆx =argmin x Ax y 2 v i double-sided exponential: p(z) =(1/2a)e z /a ML estimate is l 1 -estimate ˆx =argmin x Ax y 1 { (1/a)e z/a z 0 v i is one-sided exponential: p(z) = 0 z<0 ML estimate is found by solving LP minimize 1 T (y Ax) subject to y Ax 0 v i are uniform on [ a, a]: p(z) = { 1/(2a) a z a 0 otherwise ML estimate is any x satisfying Ax y a The linear programming problem: variants and examples 4 19

74 Linear-fractional programming c T x + d minimize f T x + g subject to Ax b f T x + g 0 (asume a/0 =+ if a>0, a/0 = if a 0) nonlinear objective function like LP, can be solved very efficiently equivalent form with linear objective (vars. x, γ): minimize γ subject to c T x + d γ(f T x + g) f T x + g 0 Ax b The linear programming problem: variants and examples 4 20

75 Bisection algorithm for linear-fractional programming given: interval [l, u] that contains optimal γ repeat: solve feasibility problem for γ =(u + l)/2 c T x + d γ(f T x + g) f T x + g 0 Ax b if feasible u := γ; if infeasible l := γ until u l ɛ each iteration is an LP feasibility problem accuracy doubles at each iteration number of iterations to reach accuracy ɛ starting with initial interval of width u l = ɛ 0 : k = log 2 (ɛ 0 /ɛ) The linear programming problem: variants and examples 4 21

76 Generalized linear-fractional programming minimize subject to max i=1,...,k c T i x + d i f T i x + g i Ax b f T i x + g i 0, i =1,...,K equivalent formulation: minimize subject to γ Ax b c T i x + d i γ(fi T x + g i), i =1,...,K fi T x + g i 0, i =1,...,K efficiently solved via bisection on γ each iteration is an LP feasibility problem The linear programming problem: variants and examples 4 22

77 Von Neumann economic growth problem simple model of an economy: m goods, n economic sectors x i (t): activity of sector i in current period t a T i x(t): amount of good i consumed in period t b T i x(t): amount of good i produced in period t choose x(t) to maximize growth rate min i x i (t +1)/x i (t): maximize γ subject to Ax(t +1) Bx(t), x(t +1) γx(t), x(t) 1 or equivalently (since a ij 0): maximize γ subject to γax(t) Bx(t), x(t) 1 (linear-fractional problem with variables x(0), γ) The linear programming problem: variants and examples 4 23

78 Optimal transmitter power allocation m transmitters, mn receivers all at same frequency transmitter i wants to transmit to n receivers labeled (i, j), j =1,...,n transmitter k receiver (i, j) transmitter i A ijk is path gain from transmitter k to receiver (i, j) N ij is (self) noise power of receiver (i, j) variables: transmitter powers p k, k =1,...,m The linear programming problem: variants and examples 4 24

79 at receiver (i, j): signal power: S ij = A iji p i noise plus interference power: I ij = k i A ijkp k + N ij signal to interference/noise ratio (SINR): S ij /I ij problem: choose p i to maximize smallest SINR: maximize subject to A iji p min i i,j k i A ijkp k + N ij 0 p i p max a (generalized) linear-fractional program special case with analytical solution: m =1, no upper bound on p i (see exercises) The linear programming problem: variants and examples 4 25

80 The linear programming problem: variants and examples 4 26

81 ESE504 (Fall 2010) Lecture 5 Structural optimization minimum weight truss design truss topology design limit analysis design with minimum number of bars 5 1

82 Truss m bars with lengths l i and cross-sectional areas x i N nodes; nodes 1,...,n are free, nodes n +1,...,N are anchored external load: forces f i R 2 at nodes i =1,...,n design problems: given the topology (i.e., location of bars and nodes), find the lightest truss that can carry a given load (vars: bar sizes x k, cost: total weight) same problem, where cost #bars used find best topology find lightest truss that can carry several given loads analysis problem: for a given truss, what is the largest load it can carry? Structural optimization 5 2

83 Material characteristics u i R is force in bar i (u i > 0: tension, u i < 0: compression) s i R is deformation of bar i (s i > 0: lengthening, s i < 0: shortening) we assume the material is rigid/perfectly plastic: u i /x i α s i s i =0 if α<u i /x i <α u i /x i = α if s i > 0 u i /x i = α if s i < 0 α (α is a material constant) Structural optimization 5 3

84 Minimum weight truss for given load force equilibrium for (free) node i: m j=1 u j [ nij,x n ij,y ] + [ fi,x f i,y ] =0 f i,y n ij bar j n ij depends on topology: n ij =0if bar j is not connected to node i node i θ ij f i,x n ij =(cosθ ij, sin θ ij ) otherwise minimum weight truss design via LP: minimize subject to m i=1 l ix i m j=1 u jn ij + f i =0, i =1,...,n αx j u j αx j, j =1,...,m (variables x j, u j ) Structural optimization 5 4

85 example bar 1 45 bar 3 node 1 45 bar 2 f mimimize l 1 x 1 + l 2 x 2 + l 3 x 3 subject to u 1 / 2 u 2 / 2 u 3 + f x =0 u 1 / 2 u 2 / 2+f y =0 αx 1 u 1 αx 1 αx 2 u 2 αx 2 αx 3 u 3 αx 3 Structural optimization 5 5

86 Truss topology design grid of nodes; bars between any pair of nodes design minimum weight truss: u i =0for most bars optimal topology: only use bars with u i 0 example: grid, i.e., 220 (potential) nodes, 24,090 (potential) bars nodes a, b, c are fixed; unit vertical force at node d optimal topology has 289 bars a bc d Structural optimization 5 6

87 Multiple loading scenarios minimum weight truss that can carry M possible loads f 1 i,...,f M i : minimize subject to m i=1 l ix i m j=1 uk j n ij + fi k =0, i =1,...,n, k =1,...,M αx j u k j αx j, j =1,...,m, k =1,...,M (variables x j, u 1 j,...,um j ) adds robustness: truss can carry any load with λ k 0, k λ k 1 f i = λ 1 f 1 i + + λ M f M i Structural optimization 5 7

88 Limit analysis truss with given geometry (including given cross-sectional areas x i ) load f i is given up to a constant multiple: f i = γg i, with given g i R 2 and γ>0 find largest load that the truss can carry: maximize subject to γ m j=1 u jn ij + γg i =0, i =1,...,n αx j u j αx j, j =1,...,m an LP in γ, u j maximum allowable γ is called the safety factor Structural optimization 5 8

89 Design with smallest number of bars integer LP formulation (assume wlog x i 1): minimize subject to m j=1 z j m j=1 u jn ij + f i =0, i =1,...,n αx j u j αx j, j =1,...,m x j z j, j =1,...,m z j {0, 1}, j =1,...,m variables z j, x j, u j extremely hard to solve; we may have to enumerate all 2 m possible values of z heuristic: replace z j {0, 1} by 0 z j 1 yields an LP; at the optimum many (but not all) z j s will be 0 or 1 called LP relaxation of the integer LP Structural optimization 5 9

90 Structural optimization 5 10

91 ESE504 (Fall 2010) Lecture 6 FIR filter design FIR filters linear phase filter design magnitude filter design equalizer design 6 1

92 FIR filters finite impulse response (FIR) filter: y(t) = n 1 τ=0 h τ u(t τ), t Z u : Z R is input signal; y : Z R is output signal h i R are called filter coefficients; n is filter order or length filter frequency response: H : R C H(ω) = h 0 + h 1 e jω + + h n 1 e j(n 1)ω = n 1 t=0 h t cos tω j n 1 t=0 h t sin tω (j = 1) periodic, conjugate symmetric, so only need to know/specify for 0 ω π FIR filter design problem: choose h so H and h satisfy/optimize specs FIR filter design 6 2

93 example: (lowpass) FIR filter, order n =21 impulse response h: 0.2 h(t) t frequency response magnitude H(ω) and phase H(ω): H(ω) H(ω) ω ω FIR filter design 6 3

94 Linear phase filters suppose n =2N +1is odd and impulse response is symmetric about midpoint: h t = h n 1 t, t =0,...,n 1 then H(ω) = h 0 + h 1 e jω + + h n 1 e j(n 1)ω = e jnω (2h 0 cos Nω +2h 1 cos(n 1)ω + + h N ) = e jnω H(ω) term e jnω represents N-sample delay H(ω) is real H(ω) = H(ω) called linear phase filter ( H(ω) is linear except for jumps of ±π) FIR filter design 6 4

95 Lowpass filter specifications δ 1 1/δ 1 δ 2 ωp ωs π specifications: maximum passband ripple (±20 log 10 δ 1 in db): ω 1/δ 1 H(ω) δ 1, 0 ω ω p minimum stopband attenuation ( 20 log 10 δ 2 in db): H(ω) δ 2, ω s ω π FIR filter design 6 5

96 Linear phase lowpass filter design sample frequency (ω k = kπ/k, k =1,...,K) can assume wlog H(0) > 0, so ripple spec is 1/δ 1 H(ω k ) δ 1 design for maximum stopband attenuation: minimize δ 2 subject to 1/δ 1 H(ω k ) δ 1, 0 ω k ω p δ 2 H(ω k ) δ 2, ω s ω k π passband ripple δ 1 is given an LP in variables h, δ 2 known (and used) since 1960 s can add other constraints, e.g., h i α FIR filter design 6 6

97 example linear phase filter, n =31 passband [0, 0.12π]; stopband [0.24π, π] max ripple δ 1 =1.059 (±0.5dB) design for maximum stopband attenuation impulse response h and frequency response magnitude H(ω) h(t) t H(ω) ω FIR filter design 6 7

98 Some variations H(ω) =2h 0 cos Nω +2h 1 cos(n 1)ω + + h N minimize passband ripple (given δ 2, ω s, ω p, N) minimize δ 1 subject to 1/δ 1 H(ω k ) δ 1, 0 ω k ω p δ 2 H(ω k ) δ 2, ω s ω k π minimize transition bandwidth (given δ 1,δ 2,ω p,n) minimize ω s subject to 1/δ 1 H(ω k ) δ 1, 0 ω k ω p δ 2 H(ω k ) δ 2, ω s ω k π FIR filter design 6 8

99 minimize filter order (given δ 1, δ 2, ω s, ω p ) minimize N subject to 1/δ 1 H(ω k ) δ 1, 0 ω k ω p δ 2 H(ω k ) δ 2, ω s ω k π can be solved using bisection each iteration is an LP feasibility problem FIR filter design 6 9

100 Filter magnitude specifications transfer function magnitude spec has form L(ω) H(ω) U(ω), ω [0,π] where L, U : R R + are given and H(ω) = n 1 t=0 h t cos tω j n 1 t=0 h t sin tω arises in many applications, e.g., audio, spectrum shaping not equivalent to a set of linear inequalities in h (lower bound is not even convex) can change variables and convert to set of linear inequalities FIR filter design 6 10

101 Autocorrelation coefficients autocorrelation coefficients associated with impulse response h =(h 0,...,h n 1 ) R n are r t = n 1 t τ=0 h τ h τ+t (with h k =0for k<0 or k n) r t = r t and r t =0for t n; hence suffices to specify r =(r 0,...,r n 1 ) Fourier transform of autocorrelation coefficients is R(ω) = τ e jωτ r τ = r 0 + n 1 t=1 2r t cos ωt = H(ω) 2 can express magnitude specification as L(ω) 2 R(ω) U(ω) 2, ω [0,π]... linear inequalities in r FIR filter design 6 11

102 Spectral factorization question: when is r R n the autocorrelation coefficients of some h R n? answer (spectral factorization theorem): if and only if R(ω) 0 for all ω spectral factorization condition is convex in r (a linear inequality for each ω) many algorithms for spectral factorization, i.e., finding an h such that R(ω) = H(ω) 2 magnitude design via autocorrelation coefficients: use r as variable (instead of h) add spectral factorization condition R(ω) 0 for all ω optimize over r use spectral factorization to recover h FIR filter design 6 12

103 Magnitude lowpass filter design maximum stopband attenuation design with variables r becomes minimize δ2 subject to 1/ δ 1 R(ω) δ 1, ω [0,ω p ] R(ω) δ 2, ω [ω s,π] R(ω) 0, ω [0,π] ( δ i corresponds to δ 2 i in original problem) now discretize frequency: minimize δ2 subject to 1/ δ 1 R(ω k ) δ 1, 0 ω k ω p R(ω k ) δ 2, ω s ω k π R(ω k ) 0, 0 ω k π...anlpinr, δ 2 FIR filter design 6 13

104 Equalizer design g(t) h(t) (time-domain) equalization: given g (unequalized impulse response) g des (desired impulse response) design (FIR equalizer) h so that g = h g g des common choice: pure delay D: g des (t) = { 1 t = D 0 t D as an LP: minimize max t D g(t) subject to g(d) =1 FIR filter design 6 14

105 example unequalized system G is 10th order FIR: g(t) t G(ω) 10 0 G(ω) ω ω FIR filter design 6 15

106 design 30th order FIR equalizer with G(ω) e j10ω minimize max t 10 g(t) equalized system impulse response g g(t) t equalized frequency response magnitude G and phase G G(ω) 10 0 G(ω) ω ω FIR filter design 6 16

107 Magnitude equalizer design H(ω) G(ω) given system frequency response G :[0,π] C design FIR equalizer H so that G(ω)H(ω) 1: minimize max ω [0,π] G(ω)H(ω) 2 1 use autocorrelation coefficients as variables: minimize α subject to G(ω) 2 R(ω) 1 α, ω [0,π] R(ω) 0, ω [0,π] when discretized, an LP in r, α,... FIR filter design 6 17

108 Multi-system magnitude equalization given M frequency responses G k :[0,π] C design FIR equalizer H so that G k (ω)h(ω) constant: minimize max k=1,...,m max ω [0,π] G k (ω)h(ω) 2 γ k subject to γ k 1, k =1,...,M use autocorrelation coefficients as variables: minimize α subject to G k (ω) 2 R(ω) γ k α, ω [0,π], k =1,...,M R(ω) 0, ω [0,π] γ k 1, k =1,...,M... when discretized, an LP in γ k, r, α FIR filter design 6 18

109 example. M =2, n =25, γ k 1 unequalized and equalized frequency responses Gk(ω) Gk(ω)H(ω) ω ω FIR filter design 6 19

110 FIR filter design 6 20

111 ESE504 (Fall 2010) Lecture 7 Applications in control optimal input design robust optimal input design pole placement (with low-authority control) 7 1

112 Linear dynamical system y(t) =h 0 u(t)+h 1 u(t 1) + h 2 u(t 2) + single input/single output: input u(t) R, output y(t) R h i are called impulse response coefficients finite impulse response (FIR) system of order k: h i =0for i>k if u(t) =0for t<0, y(0) y(1) y(2). y(n) = h h 1 h h 2 h 1 h h N h N 1 h N 2 h 0 u(0) u(1) u(2). u(n) a linear mapping from input to output sequence Applications in control 7 2

113 Output tracking problem choose inputs u(t), t =0,...,M (M <N)that minimize peak deviation between y(t) and a desired output y des (t), t =0,...,N, max t=0,...,n y(t) y des(t) satisfy amplitude and slew rate constraints: u(t) U, u(t +1) u(t) S as a linear program (variables: w, u(0),...,u(n)): minimize. subject to w w t i=0 h iu(t i) y des (t) w, t =0,...,N u(t) =0, t = M +1,...,N U u(t) U, t =0,...,M S u(t +1) u(t) S, t =0,...,M +1 Applications in control 7 3

114 example. single input/output, N = 200 step response y des constraints on u: input horizon M = 150 amplitude constraint u(t) 1.1 slew rate constraint u(t) u(t 1) 0.25 Applications in control 7 4

115 output and desired output: y(t), y des (t) optimal input sequence u: 1.1 u(t) 0.25 u(t) u(t 1) Applications in control 7 5

116 Robust output tracking (1) impulse response is not exactly known; it can take two values: (h (1) 0,h(1) 1,...,h(1) k ), (h(2) 0,h(2) 1,...,h(2) k ) design an input sequence that minimizes the worst-case peak tracking error minimize w subject to w t i=0 h(1) i u(t i) y des (t) w, t =0,...,N w t i=0 h(2) i u(t i) y des (t) w, t =0,...,N u(t) =0, t = M +1,...,N U u(t) U, t =0,...,M S u(t +1) u(t) S, t =0,...,M +1 an LP in the variables w, u(0),...,u(n) Applications in control 7 6

117 example step responses outputs and desired output u(t) 0.25 u(t) u(t 1) Applications in control 7 7

118 Robust output tracking (2) h i and v (j) i h 0 (s) h 1 (s). h k (s) = h 0 h 1. h k + s 1 v (1) 0 v (1) 1. v (1) k are given; s i [ 1, +1] is unknown + + s K v (K) 0 v (K) 1. v (K) k robust output tracking problem (variables w, u(t)): min. s.t. w w t i=0 h i(s)u(t i) y des (t) w, t =0,...,N, s [ 1, 1] K u(t) =0, t = M +1,...,N U u(t) U, t =0,...,M S u(t +1) u(t) S, t =0,...,M +1 straightforward (and very inefficient) solution: enumerate all 2 K extreme values of s Applications in control 7 8

119 simplification: we can express the 2 K+1 linear inequalities w t h i (s)u(t i) y des (t) w for all s { 1, 1} K i=0 as two nonlinear inequalities t K h i u(t i)+ i=0 j=1 t i=0 i u(t i) y des(t)+w v (j) t h i u(t i) i=0 K j=1 t i=0 i u(t i) y des(t) w v (j) Applications in control 7 9

120 proof: max s { 1,1} K = = t h i (s)u(t i) i=0 t h i u(t i)+ i=0 t h i u(t i)+ i=0 K t max s j s j { 1,+1} i=0 K t v (j) i u(t i) j=1 j=1 i=0 v (j) i u(t i) and similarly for the lower bound Applications in control 7 10

121 robust output tracking problem reduces to: min. s.t. w t h i=0 i u(t i)+ K j=1 t i=0 v(j) i u(t i) y des (t)+w, t h i=0 i u(t i) K j=1 t i=0 v(j) i u(t i) y des (t) w, u(t) =0, t = M +1,...,N U u(t) U, t =0,...,M S u(t +1) u(t) S, t =0,...,M +1 t =0,...,N t =0,...,N (variables u(t), w) to express as an LP: for t =0,...,N, j =1,...,K, introduce new variables p (j) (t) and constraints t p (j) (t) v (j) i u(t i) p (j) (t) i=0 replace i v(j) i u(t i) by p (j) (t) Applications in control 7 11

122 example (K =6) nominal and perturbed step responses 1 0 design for nominal system output for nominal system output for worst-case system Applications in control 7 12

123 robust design output for nominal system output for worst-case system Applications in control 7 13

124 input-output description: State space description y(t) =H 0 u(t)+h 1 u(t 1) + H 2 u(t 2) + if u(t) =0, t<0: y(0) y(1) y(2). y(n) = H H 1 H H 2 H 1 H H N H N 1 H N 2 H 0 u(0) u(1) u(2). u(n) block Toeplitz structure (constant along diagonals) state space model: x(t +1)=Ax(t)+Bu(t), y(t) =Cx(t)+Du(t) with H 0 = D, H i = CA i 1 B (i >0) x(t) R n is state sequence Applications in control 7 14

125 alternative description: y(0) y(1). y(n) = A I 0 0 B A I 0 0 B I 0 0 B C D C D C 0 0 D x(0) x(1) x(2). x(n) u(0) u(1). u(n) we don t eliminate the intermediate variables x(t) matrix is larger, but very sparse (interesting when using general-purpose LP solvers) Applications in control 7 15

126 Pole placement linear system ż(t) =A(x)z(t), z(0) = z 0 where A(x) =A 0 + x 1 A x p A p R n n solutions have the form z i (t) = k β ik e σ kt cos(ω k t φ ik ) where λ k = σ k ± jω k are the eigenvalues of A(x) x R p is the design parameter goal: place eigenvalues of A(x) in a desired region by choosing x Applications in control 7 16

127 Low-authority control eigenvalues of A(x) are very complicated (nonlinear, nondifferentiable) functions of x first-order perturbation: if λ i (A 0 ) is simple, then λ i (A(x)) = λ i (A 0 )+ p k=1 wi A kv i wi v x k + o( x ) i where w i, v i are the left and right eigenvectors: w i A 0 = λ i (A 0 )w i, A 0 v i = λ i (A 0 )v i low-authority control: use linear first-order approximations for λ i can place λ i in a polyhedral region by imposing linear inequalities on x we expect this to work only for small shifts in eigenvalues Applications in control 7 17

128 Example trusswith30nodes,83bars M d(t)+d d(t)+kd(t) =0 d(t): vector of horizontal and vertical node displacements M = M T > 0 (mass matrix): masses at the nodes D = D T > 0 (damping matrix); K = K T > 0 (stiffness matrix) to increase damping, we attach dampers to the bars: D(x) =D 0 + x 1 D x p D p x i > 0: amount of external damping at bar i Applications in control 7 18

129 eigenvalue placement problem minimize p i=1 x i subject to λ i (M,D(x),K) C, i =1,...,n x 0 an LP if C is polyhedral and we use the 1st order approximation for λ i 5 before 4 eigenvalues 5 after Applications in control 7 19

130 location of dampers Applications in control 7 20

131 ESE504 (Fall 2010) Lecture 8 Duality (part 1) the dual of an LP in inequality form weak duality examples optimality conditions and complementary slackness Farkas lemma and theorems of alternatives proofofstrongduality 8 1

132 LP in inequality form: The dual of an LP in inequality form minimize c T x subject to a T i x b i, i =1,...,m n variables, m inequality constraints, optimal value p called primal problem (in context of duality) the dual LP (with A =[a 1 a 2... a m ] T ): maximize b T z subject to A T z + c =0 z 0 an LP in standard form with m variables, n equality constraints optimal value denoted d main property: p = d (if primal or dual is feasible) Duality (part 1) 8 2

133 Weak duality lower bound property: if x is primal feasible and z is dual feasible, then c T x b T z proof: c T x c T x + m i=1 z i(a T i x b i)= b T z c T x + b T z is called the duality gap associated with x and z weak duality: minimize over x, maximize over z: p d always true (even when p =+ and/or d = ) Duality (part 1) 8 3

134 example primal problem minimize 4x 1 5x subject to [ x1 x 2 ] optimal point: x =(1, 1), optimal value p = 9 dual problem maximize 3z 2 3z 4 subject to [ ] z 1 z 2 z 3 z 4 z 1 0, z 2 0, z 3 0, z [ 4 5 z =(0, 1, 0, 2) is dual feasible with objective value 9 ] =0 Duality (part 1) 8 4

135 conclusion (by weak duality): z is a certificate that x is (primal) optimal x is a certificate that z is (dual) optimal Duality (part 1) 8 5

136 Piecewise-linear minimization minimize max i=1,...,m (a T i x b i) lower bounds for optimal value p? LP formulation (variables x, t) minimize t subject to [ A 1 ] [ x t ] b dual LP maximize subject to (same optimal value) b T z [ A T 1 T z 0 ] z + [ 0 1 ] =0 Duality (part 1) 8 6

137 Interpretation lemma: if z 0, i z i =1,thenforally, max i y i i z i y i hence, max i (a T i x b i) z T (Ax b) this yields a lower bound on p : p =min x max(a T i x b i ) min z T (Ax b) = i x { b T z if A T z =0 otherwise to get best lower bound: maximize b T z subject to A T z =0 1 T z =1 z 0 Duality (part 1) 8 7

138 l -approximation LP formulation minimize Ax b minimize subject to [ t A A 1 1 ][ x t ] [ b b ] LP dual canbeexpressedas maximize b T w + b T v subject to A T w A T v =0 1 T w + 1 T v =1 w, v 0 maximize b T z subject to A T z =0 z 1 1 (1) (2) Duality (part 1) 8 8

139 proofofequivalenceof(1)and(2) assume w, v feasible in (1), i.e., w 0, v 0, 1 T (w + v) =1 z = v w is feasible in (2): z 1 = i v i w i 1 T v + 1 T w =1 same objective value: b T z = b T v b T w assume z is feasible in (2), i.e., A T z =0, z 1 1 w i =max{z i, 0} + α, v i =max{ z i, 0} + α, with α =(1 z 1 )/(2m), are feasible in (1): v, w 0, 1 T w + 1 T v =1 same objective value: b T v b T w = b T z Duality (part 1) 8 9

140 Interpretation lemma: u T v u 1 v hence, for every z with z 1 1, wehavealowerboundon Ax b : Ax b z T (Ax b) p =min x togetbestlowerbound Ax b min x z T (Ax b) = maximize b T z subject to A T z =0 z 1 1 { b T z if A T z =0 otherwise Duality (part 1) 8 10

141 Optimality conditions primal feasible x is optimal if and only if there is a dual feasible z with c T x = b T z i.e., associated duality gap is zero complementary slackness: forx, z optimal, c T x + b T z = m z i (b i a T i x) =0 i=1 hence for each i, a T i x = b i or z i =0: z i > 0= a T i x = b i (ith inequality is active at x) a T i x<b i = z i =0 Duality (part 1) 8 11

142 Geometric interpretation example in R 2 : a 1 a 1 c a 2 a 2 two active constraints at optimum (a T 1 x = b 1, a T 2 x = b 2 ) optimal dual solution satisfies c = A T z, z 0, z i =0for i 1, 2, i.e., c = a 1 z 1 + a 2 z 2 geometrically, c lies in the cone generated by a 1 and a 2 Duality (part 1) 8 12

143 Separating hyperplane theorem if S R n is a nonempty, closed, convex set, and x S, then there exists c 0such that c T x <c T x for all x S, i.e., for some value of d, the hyperplane c T x = d separates x from S x c p S (x ) S idea of proof: use c = p S (x ) x,wherep S (x ) is the projection of x on S, i.e., p S (x ) = argmin x x x S Duality (part 1) 8 13

144 Farkas lemma given A, b, exactly one of the following two statements is true: 1. there is an x 0 such that Ax = b 2. there is a y such that A T y 0, b T y<0 very useful in practice: any y in 2 is a certificate or proof that Ax = b, x 0 is infeasible, and vice-versa proof (easy part): we have a contradiction if 1 and 2 are both true: 0=y T (Ax b) b T y>0 Duality (part 1) 8 14

145 proof (difficult part): 1 = 2 1 means b S = {Ax x 0} S is nonempty, closed, and convex (the image of the nonnegative orthant under a linear mapping) hence there exists a y s.t. implies: y T b<y T Ax for all x 0 y T b<0 (choose x =0) A T y 0 (if (A T y) k < 0 for some k, wecanchoosex i =0for i k, and x k + ; theny T Ax ) i.e., 2istrue Duality (part 1) 8 15

146 Theorems of alternatives many variations on Farkas lemma: e.g., forgivena R m n b R m, exactly one of the following statements is true: 1. there is an x with Ax b 2. there is a y 0 with A T y =0, b T y<0 proof (easy half): 1 and 2 together imply 0 (b Ax) T y = b T y<0 (difficult half): if 1 does not hold, then b S = {Ax + s x R n,s R m,s 0} hence, there is a separating hyperplane, i.e., y 0subject to y T b<y T (Ax + s) for all x and all s 0 equivalent to b T y<0, A T y =0, y 0 (i.e., 2istrue) Duality (part 1) 8 16

147 Proof of strong duality strong duality: p = d (except possibly when p =+, d = ) suppose p is finite, and x is optimal with a T i x = b i, i I, a T i x <b i, i I we ll show there is a dual feasible z with b T z = c T x x optimal implies that the set of inequalities a T i d 0, i I, c T d<0 (1) is infeasible; otherwise we would have for small t>0 a T i (x + td) b i, i =1,...,m, c T (x + td) <c T x Duality (part 1) 8 17

148 from Farkas lemma: (1) is infeasible if and only if there exists λ i, i I, λ i 0, λ i a i = c i I this yields a dual feasible z: z i = λ i, i I, z i =0, i I z is dual optimal: b T z = i I b i z i = i I (a T i x )z i = z T Ax = c T x this proves: p finite = d = p exercise: p =+ = d =+ or d = Duality (part 1) 8 18

149 Summary possible cases: p = d and finite: primal and dual optima are attained p = d =+ : primal is infeasible; dual is feasible and unbounded p = d = : primal is feasible and unbounded; dual is infeasible p =+, d = : primal and dual are infeasible uses of duality: dual optimal z provides a proof of optimality for primal feasible x dual feasible z provides a lower bound on p (useful for stopping criteria) sometimes it is easier to solve the dual modern interior-point methods solve primal and dual simultaneously Duality (part 1) 8 19

150 Duality (part 1) 8 20

151 ESE504 (Fall 2010) Lecture 9 Duality (part 2) duality in algorithms sensitivity analysis via duality duality for general LPs examples mechanics interpretation circuits interpretation two-person zero-sum games 9 1