Lecture 1 Introduction


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1 L. Vandenberghe EE236A (Fall ) Lecture 1 Introduction course overview linear optimization examples history approximate syllabus basic definitions linear optimization in vector and matrix notation halfspaces and polyhedra geometrical interpretation 1 1
2 Linear optimization minimize subject to n j=1 n j=1 n j=1 c j x j a ij x j b i, i = 1,...,m d ij x j = f i, i = 1,...,p n optimization variables: x 1,..., x n (real scalars) problem data (parameters): the coefficients c j, a ij, b i, d ij, f i j c jx j is the cost function or objective function j a ijx j b i and j d ijx j = f i are inequality and equality constraints called a linear optimization problem or linear program (LP) Introduction 1 2
3 Importance low complexity problems with several thousand variables, constraints routinely solved much larger problems (millions of variables) if problem data are sparse widely available software theoretical worstcase complexity is polynomial wide applicability originally developed for applications in economics and management today, used in all areas of engineering, data analysis, finance,... a key tool in combinatorial optimization extensive theory no simple formula for solution but extensive, useful (duality) theory Introduction 1 3
4 Example: openloop control problem singleinput/singleoutput system (input u(t), output y(t) at time t) 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 y(t) y des(t) t=0,...,n 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) = 0 for t < 0, t > M) solution: can be formulated as an LP, hence easily solved (more later) Introduction 1 4
5 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 1 5
6 optimal solution (computed via linear optimization) 1.1 input u(t) 0.25 u(t) u(t 1) output and desired output Introduction 1 6
7 Example: assignment problem match N people to N tasks each person is assigned to one task; each task assigned to one person cost of assigning person i to task j is a ij combinatorial formulation minimize subject to N i,j=1 N i=1 N j=1 a ij x ij x ij = 1, j = 1,...,N x ij = 1, i = 1,...,N x ij {0,1}, i,j = 1,...,N variable x ij = 1 if person i is assigned to task j; x ij = 0 otherwise N! possible assignments, i.e., too many to enumerate Introduction 1 7
8 linear optimization formulation minimize subject to N i,j=1 N i=1 N j=1 a ij x ij x ij = 1, j = 1,...,N x ij = 1, i = 1,...,N 0 x ij 1, i,j = 1,...,N we have relaxed the constraints x ij {0,1} it can be shown that at the optimum x ij {0,1} (see later) hence, can solve (this particular) combinatorial problem efficiently (via linear optimization or specialized methods) Introduction 1 8
9 Brief history 1940s (Dantzig, Kantorovich, Koopmans, von Neumann,... ) foundations, motivated by economics and logistics problems 1947 (Dantzig): simplex algorithm 1950s 60s: applications in other disciplines 1979 (Khachiyan): ellipsoid algorithm: more efficient (polynomialtime) than simplex in worst case, much slower in practice 1984 (Karmarkar): projective (interiorpoint) algorithm: polynomialtime worstcase complexity, and efficient in practice since 1984: variations of interiorpoint methods (improved complexity or efficiency in practice), software for largescale problems Introduction 1 9
10 Tentative syllabus linear and piecewiselinear optimization polyhedral geometry duality applications algorithms: simplex algorithm, interiorpoint algorithms, decomposition applications in network and combinatorial optimization extensions: linearfractional programming introduction to integer linear programming Introduction 1 10
11 Vectors vector of length n (or nvector) x = x 1 x 2. x n we also use the notation x = (x 1,x 2,...,x n ) x i is ith component or element (real unless specified otherwise) set of real nvectors is denoted R n special vectors (with n determined from context) x = 0 (zero vector): x i = 0, i = 1,...,n x = 1 (vector of all ones): x i = 1, i = 1,...,n x = e i (ith basis or unit vector): x i = 1, x k = 0 for k i Introduction 1 11
12 Matrices matrix of size m n A = A 11 A 12 A 1n A 21. A 22. A 2n. A m1 A m2 A mn A ij (or a ij ) is the i,j element (or entry, coefficient) set of real m nmatrices is denoted R m n vectors can be viewed as matrices with one column special matrices (with size determined from context) X = 0 (zero matrix): X ij = 0 for i = 1,...,m, j = 1,...,n X = I (identity matrix): m = n with X ii = 1, X ij = 0 for i j Introduction 1 12
13 Operations matrix transpose A T scalar multiplication αa addition A+B and subtraction A B of matrices of the same size product y = Ax of a matrix with a vector of compatible length product C = AB of matrices of compatible size inner product of nvectors: x T y = x 1 y 1 + +x n y n Introduction 1 13
14 LP in innerproduct notation minimize subject to n j=1 n j=1 n j=1 c j x j a ij x j b i, i = 1,...,m d ij x j = f i, i = 1,...,p innerproduct notation c, a i, d i are nvectors: minimize c T x subject to a T i x b i, i = 1,...,m d T i x = f i, i = 1,...,p c = (c 1,...,c n ), a i = (a i1,...,a in ), d i = (d i1,...,d in ) Introduction 1 14
15 LP in matrix notation matrix notation minimize subject to n j=1 n j=1 n j=1 c j x j a ij x j b i, i = 1,...,m d ij x j = f i, i = 1,...,p minimize c T x subject to Ax b Dx = f A is m nmatrix with elements a ij, rows a T i D is p nmatrix with elements d ij, rows d T i inequality is componentwise vector inequality Introduction 1 15
16 Terminology minimize c T x subject to Ax b Dx = f x is feasible if it satisfies the constraints Ax b and Dx = f feasible set is set of all feasible points x is optimal if it is feasible and c T x c T x for all feasible x the optimal value of the LP is p = c T x unbounded problem: c T x unbounded below on feasible set (p = ) infeasible probem: feasible set is empty (p = + ) Introduction 1 16
17 Euclidean norm x = Vector norms x 2 1 +x x2 n = x T x l 1 norm and l norm x 1 = x 1 + x x n x = max{ x 1, x 2,..., x n } properties (satisfied by any norm f(x)) f(αx) = α f(x) (homogeneity) f(x+y) f(x)+f(y) (triangle inequality) f(x) 0 (nonnegativity); f(x) = 0 if only if x = 0 (definiteness) Introduction 1 17
18 CauchySchwarz inequality x y x T y x y holds for all vectors x, y of the same size x T y = x y iff x and y are aligned (nonnegative multiples) x T y = x y iff x and y are opposed (nonpositive multiples) implies many useful inequalities as special cases, for example, n x n x i n x i=1 Introduction 1 18
19 Angle between vectors the angle θ = (x,y) between nonzero vectors x and y is defined as θ = arccos x T y x y (i.e., x T y = x y cosθ) we normalize θ so that 0 θ π relation between sign of inner product and angle x T y > 0 θ < π 2 x T y = 0 θ = π 2 x T y < 0 θ > π 2 (vectors make an acute angle) (orthogonal vectors) (vectors make an obtuse angle) Introduction 1 19
20 Projection projection of x on the line defined by nonzero y: the vector ˆty with ˆt = argmin t x ty expression for ˆt: ˆt = xt y y 2 = x cosθ y ( ) ˆty = x T y y y T y y x 0 Introduction 1 20
21 Hyperplanes and halfspaces hyperplane solution set of one linear equation with nonzero coefficient vector a a T x = b halfspace solution set of one linear inequality with nonzero coefficient vector a a T x b a is the normal vector Introduction 1 21
22 Geometrical interpretation G = {x a T x = b} H = {x a T x b} a a u = (b/ a 2 )a x G x u 0 x u x u H the vector u = (b/ a 2 )a satisfies a T u = b x is in hyperplane G if a T (x u) = 0 (x u is orthogonal to a) x is in halfspace H if a T (x u) 0 (angle (x u,a) π/2) Introduction 1 22
23 Example a T x = 0 x 2 a T x = 10 x 2 a = (2,1) a = (2,1) x 1 x 1 a T x = 5 a T x = 5 a T x 3 Introduction 1 23
24 Polyhedron solution set of a finite number of linear inequalities a T 1x b 1, a T 2x b 2,..., a T mx b m a 1 a 2 a 5 a 3 a 4 intersection of a finite number of halfspaces in matrix notation: Ax b if A is a matrix with rows a T i can include equalities: Fx = g is equivalent to Fx g, Fx g Introduction 1 24
25 Example x 1 +x 2 1, 2x 1 +x 2 2, x 1 0, x 2 0 x 2 2x 1 +x 2 = 2 x 1 +x 2 = 1 x 1 Introduction 1 25
26 Example 0 x 1 2, 0 x 2 2, 0 x 3 2, x 1 +x 2 +x 3 5 x 3 (0,0,2) (0,2,2) (1,2,2) (2,0,2) (2,1,2) (2,2,1) (0,2,0) x 2 (2,0,0) (2,2,0) x 1 Introduction 1 26
27 Geometrical interpretation of LP minimize c T x subject to Ax b c Ax b optimal solution dashed lines (hyperplanes) are level sets c T x = α for different α Introduction 1 27
28 Example minimize x 1 x 2 subject to 2x 1 +x 2 3 x 1 +4x 2 5 x 1 0, x 2 0 x 1 x 2 = 2 x 2 x 1 x 2 = 1 x 1 x 2 = 5 x 1 x 2 = 0 x 1 x 2 = 4 x 1 optimal solution is (1, 1) x 1 x 2 = 3 Introduction 1 28
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