1 / 44

IE 531 Linear Programming

IE 531 Linear Programming. Spring 2011. 박 성 수. Course Objectives. Why need to study LP? Important tool by itself Theoretical basis for later developments (IP, Network, Graph, Nonlinear, scheduling, Sets, Coding, Game, … )

iokina
Download Presentation

IE 531 Linear Programming

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. IE 531 Linear Programming Spring 2011 박 성 수

  2. Course Objectives • Why need to study LP? • Important tool by itself • Theoretical basis for later developments (IP, Network, Graph, Nonlinear, scheduling, Sets, Coding, Game, … ) • Formulation + package is not enough for advanced applications and interpretation of results • Objectives of the class: • Understand the theory of linear optimization • Preparation for more in-depth optimization theory • Modeling capabilities • Introduction to use of software (Xpress-MP and/or CPLEX)

  3. Prerequisite: basic linear algebra/calculus, earlier exposure to LP/OR helpful, mathematical maturity (reading proofs, logical thinking) • Be steady in studying.

  4. Chapter 1 Introduction • Mathematical Programming Problem: min/max f(x) subject to gi(x)  0, i = 1, ..., m, (hj(x) = 0, j = 1, ..., k,) ( x  X  Rn) f, gi, hj : Rn  R • If f, gi, hj linear (affine) function  linear programming problem If f, gi, hj (or part of them) nonlinear function  nonlinear programming problem If solution set restricted to be integer points  integer programming problem

  5. Linear programming: problem of optimizing (maximize or minimize) a linear (objective) function subject to linear inequality (and equality) constraints. • General form: {max, min} c'x subject to ai'x  bi , iM1 ai'x  bi , iM2 ai'x = bi , iM3 xj  0, jN1 , xj  0, jN2 c, ai , x Rn (There may exist variables unrestricted in sign) • inner product of two column vectors x, y  Rn : x’y = i = 1n xiyi If x’y = 0, x, y  0, then x, y are said to be orthogonal. In 3-D, the angle between the two vectors is 90 degrees. ( vectors are column vectors unless specified otherwise)

  6. Big difference from systems of linear equations is the existence of objective function and linear inequalities (instead of equalities) • Much deeper theoretical results and applicability than systems of linear equations. • x1, x2, …, xn : (decision) variables bi : right-hand-side ai'x { , ,  } bi : i th constraint xj { ,  } 0 : nonnegativity (nonpositivity) constraint c'x : objective function • Other terminology: feasible solution, feasible set (region), free (unrestricted) variable, optimal (feasible) solution, optimal cost, unbounded

  7. Important submatrix multiplications • Interpretation of constraints: see as submatrix multiplication. A: mn matrix , where ei is i-th unit vector denote constraints as Ax { , ,  } b

  8. Any LP can be expressed as min c'x, Ax  b max c'x  min (-c'x) and take negative of the optimal cost ai'x  bi  -ai'x  -bi ai'x = bi  ai'x  bi , -ai'x  -bi nonnegativity (nonpositivity) are special cases of inequalities which will be handled separately in the algorithms. Feasible solution set of LP can always be expressed as Ax  b (or Ax  b) (called polyhedron, a set which can be described as a solution set of finitely many linear inequalities) • We may sometimes use max c'x, Ax  b form (especially, when we study polyhedron)

  9. Brief History of LP (or Optimization) • Gauss: Gaussian elimination to solve systems of equations Fourier(early 19C) and Motzkin(20C) : solving systems of linear inequalities Farkas, Minkowski, Weyl, Caratheodory, … (19-20C): Mathematical structures related to LP (polyhedron, systems of alternatives, polarity) Kantorovich (1930s) : efficient allocation of resources (Nobel prize in 1975 with Koopmans) Dantzig (1947) : Simplex method 1950s : emergence of Network theory, Integer and combinatorial optimization, development of computer 1960s : more developments 1970s : disappointment, NP-completeness, more realistic expectations Khachian (1979) : ellipsoid method for LP

  10. 1980s : personal computer, easy access to data, willingness to use models Karmarkar (1984) : Interior point method 1990s : improved theory and software, powerful computers software add-ins to spreadsheets, modeling languages, large scale optimization, more intermixing of O.R. and A.I. Markowitz (1990) : Nobel prize for portfolio selection (quadratic programming) Nash (1994) : Nobel prize for game theory 21C (?) : Lots of opportunities more accurate and timely data available more theoretical developments better software and computer need for more automated decision making for complex systems need for coordination for efficient use of resources (e.g. supply chain management, APS, traditional engineering problems, bio, finance, ...)

  11. Application Areas of Optimization • Operations Managements Production Planning Scheduling (production, personnel, ..) Transportation Planning, Logistics Energy Military Finance Marketing E-business Telecommunications Games Engineering Optimization (mechanical, electrical, bioinformatics, ... ) System Design …

  12. Resources • Societies: • INFORMS (the Institute for Operations Research and Management Sciences) : www.informs.org • MPS (The Mathematical Programming Society) : www.mathprog.org • Korean Institute of Industrial Engineers : kiie.org • Korean Operations Research Society : www.korms.or.kr • Journals: Operations Research, Management Science, Mathematical Programming, Networks, European Journal of Operational Research, Naval Research Logistics, Journal of the Operational Research Society, Interfaces, …

  13. Standard form problems • Standard form : min c'x, Ax = b, x  0 Two view points: • Find optimal weights (nonnegative) from possible nonnegative linear combinations of columns of A to obtain b vector • Find optimal solution that satisfies linear equations and nonnegativity • Reduction to standard form Free (unrestricted) variable xj xj+ - xj- , xj+, xj- 0 j aijxij  bi  j aijxij + si = bi , si  0 (slack variable) j aijxij  bi  j aijxij - si = bi , si  0 (surplus variable)

  14. Any (practical) algorithm can solve the LP problem in equality form only (except nonnegativity) • Modified form of the simplex method can solve the problem with free variables directly (w/o using difference of two variables). It gives more sensible interpretation of the behavior of the algorithm.

  15. 1.2 Formulation examples • See other examples in the text. • Minimum cost network flow problem Directed network G=(N, A), (|N| = n ) arc capacity uij , (i, j) A, unit flow cost cij , (i, j) A bi : net supply at node i (bi > 0: supply node, bi < 0: demand node), (i bi = 0) Find min cost transportation plan that satisfies supply, demand at each node and arc capacities. minimize (i, j)A cijxij subject to {j : (i, j)A} xij - {j : (j, i)A} xji = bi , i = 1, …, n (out flow - in flow = net flow at node i) (some people use, in flow – out flow = net flow) xij  uij , (i, j)A xij  0 , (i, j)A

  16. Choosing paths in a communication network ( (fractional) multicommodity flow problem) • Multicommodity flow problem: Several commodities share the network. For each commodity, it is min cost network flow problem. But the commodities must share the capacities of the arcs. Generalization of min cost network flow problem. Many applications in communication, distribution / transportation systems • Several commodities case • Actually one commodity. But there are multiple origin and destination pairs of nodes (telecom, logistics, ..) • Given telecommunication network (directed) with arc set A, arc capacity uij bits/sec, (i, j) A, unit flow cost cij /bit , (i, j) A, demand bkl bits/sec for traffic from node k to node l. Data can be sent using more than one path. Find paths to direct demands with min cost.

  17. Decision variables: xijkl : amount of data with origin k and destination l that traverses link (i, j) A Let bikl = bkl if i = k -bkl if i = l 0 otherwise • Formulation (flow formulation) minimize (i, j)A k l cijxijkl subject to {j : (i, j)A} xijkl -  {j : (j, i)A} xjikl = bikl , i, k, l = 1, …, n (out flow - in flow = net flow at node i for commodity from node k to node l) k l xijkl  uij , (i, j)A (The sum of all commodities should not exceed the capacity of link (i, j) ) xijkl  0 , (i, j)A, k, l =1, …, n

  18. Alternative formulation (path formulation) Let K: set of origin-destination pairs (commodities) P(k): set of all possible paths for sending commodity k  K P(k;e): set of paths in P(k) that traverses arc e  A E(p): set of links contained in path p Decision variables: ypk : fraction of commodity k sent on path p minimize kK pP(k) wpkypk subject to pP(k) ypk = 1, for all kK kK pP(k; e) bkypk  ue , for all eA 0  ypk  1, for all p  P(k), k  K where wpk = bkeE(p) ce • If ypk  {0, 1}, it is a single path routing problem (path selection, integer multicommodity flow).

  19. path formulation has smaller number of constraints, but enormous number of variables. can be solved easily by column generation technique (later). Integer version is more difficult to solve. • Extensions: Network design - also determine the number and type of facilities to be installed on the links (and/or nodes) together with routing of traffic. • Variations: Integer flow. Bifurcation of traffic may not be allowed. Determine capacities and routing considering rerouting of traffic in case of network failure, Robust network design (data uncertainty), ...

  20. Pattern classification (Linear classifier) Given m objects with feature vector aiRn , i = 1, …, m. Objects belong to one of two classes. We know the class to which each sample object belongs. We want to design a criterion to determine the class of a new object using the feature vector. Want to find a vector (x, xn+1) Rn+1 with x Rn such that, if i S, then ai'x  xn+1, and if i S, then ai'x < xn+1. (if it is possible)

  21. Find a feasible solution (x, xn+1) that satisfies ai'x  xn+1, i S ai'x < xn+1. i S for all sample objects i Is this a linear programming problem? ( no objective function, strict inequality in constraints)

  22. Is strict inequality allowed in LP? consider min x, x > 0  no minimum point. only infimum of objective value exists • If the system has a feasible solution (x, xn+1), we can make the difference of the rhs and lhs large by using solution M(x, xn+1) for M > 0 and large. Hence there exists a solution that makes the difference at least 1 if the system has a solution. Remedy: Use ai'x  xn+1, i S ai'x  xn+1-1, i S • Important problem in data mining with applications in target marketing, bankruptcy prediction, medical diagnosis, process monitoring, …

  23. Variations • What if there are many choices of hyperplanes? any reasonable criteria? • What if there is no hyperplane separating the two classes? • Do we have to use only one hyperplane? • Use of nonlinear function possible? How to solve them? • SVM (support vector machine), convex optimization • More than two classes?

  24. 1.3 Piecewise linear convex obj. functions • Some problems involving nonlinear functions can be modeled as LP. • Def: Function f : Rn R is called a convex function if for all x, y Rn and all   [0, 1] f(x + (1- )y)  f(x) + (1- )f(y). ( the domain may be restricted) f called concave if -f is convex (picture: the line segment joining (x, f(x)) and (y, f(y)) in Rn+1 is not below the locus of f(x) )

  25. Def: x, y Rn, 1, 2  0, 1+ 2 = 1 Then 1x + 2y is said to be a convex combination of x, y. Generally, i=1k ixi , where i=1k i = 1 and i  0, i = 1, ..., k is a convex combination of the points x1, ..., 타 • Def: A set S  Rn is convex if for any x, y S, we have 1x + 2y S for any 1, 2  0, 1+ 2 = 1. Picture: 1x + 2y = 1x + (1 - 1)y, 0  1  1 = y + 1 (x – y), 0  1  1 (line segment joining x, y lies in S) x (1 = 1) (x-y) y (1 = 0) (x-y)

  26. If we have 1x + 2y, 1+ 2 = 1 (without 1, 2  0), it is called an affine combination of x and y. Picture: 1x + 2y = 1x + (1 - 1)y, = y + 1 (x – y), (1 is arbitrary) (line passing through x, y)

  27. Picture of convex function

  28. relation between convex function and convex set • Def: f: Rn R. Define epigraph of f as epi(f) = { (x, ) Rn+1 :   f(x) } • Then previous definition of convex function is equivalent to epi(f) being a convex set. When dealing with convex functions, we frequently consider epi(f) to exploit the properties of convex sets. • Consider operations on functions that preserve convexity and operations on sets that preserve convexity.

  29. Example: Consider f(x) = max i = 1, …, m (ci'x + di), ciRn, di R (maximum of affine functions, called a piecewise linear convex function.) c1'x+d1 c2'x+d2 c3'x+d3 x

  30. Thm: Let f1, …, fm : Rn R be convex functions. Then f(x) = max i = 1, …, m fi(x) is also convex. pf) f(x + (1- )y) = max i=1, …, m fi(x + (1- )y )  max i=1, …, m (fi(x) + (1- )fi(y) )  max i=1, …, m fi(x) + max i=1, …, m (1- )fi(y) = f(x) + (1- )f(y)

  31. Min of piecewise linear convex functions Minimize max I=1, …, m (ci'x + di) Subject to Ax  b Minimize z Subject to z  ci'x + di , i = 1, …, m Ax  b

  32. Q: What can we do about finding max of a piecewise linear convex function? maximum of a piecewise linear concave function (can be obtained as min of affine functions)? Min of a piecewise linear concave function?

  33. Convex function has a nice property such that a local min point is a global min point. (when domain is Rn or convex set) (HW later) Hence finding min of a convex function defined over a convex set is usually easy. But finding a max of a convex function is difficult to solve. Basically, we need to examine all local max points. Similarly, finding a max of concave function is easy, but finding min of a concave function is difficult.

  34. In constraints, f(x)  h where f(x) is piecewise linear convex function f(x) = max i=1, …, m (fi'x + gi).  fi'x + gi h, i = 1, … , m Q: What about constraints f(x)  h ? Can it be modeled as LP? • Def: f: Rn R, convex function,   R The set C = { x: f(x)   } is called the level set of f • level set of a convex function is a convex set. (HW later) solution set of LP is convex (easy)  non-convex solution set can’t be modeled as LP.

  35. Problems involving absolute values • Minimize i = 1, …, n ci |xi| subject to Ax  b (assume ci  0) More direct formulations than piecewise linear convex function is possible. (1) Min ici zi subject to Ax  b xi  zi , i = 1, …, n -xi  zi , i = 1, …, n (2) Min ici (xi+ + xi-) subject to Ax+ - Ax-  b x+ , x-  0 (want xi+ = xiif xi  0, xi- = -xiif xi < 0 and xi+xi- = 0, i.e., at most one of xi+, xi- is positive in an optimal solution. ci  0 guarantees that.)

  36. Data Fitting • Regression analysis using absolute value function Given m data points (ai , bi ), i = 1, …, m, aiRn , bi R. Want to find x Rn that predicts results b given a with function b = a'x Want x that minimizes prediction error | bi - ai'x | for all i. minimize z subject to bi - ai'x  z, i = 1, … , m -bi + ai'x  z, i = 1, … , m

  37. Alternative criterion minimize i = 1, …, m | bi - ai'x | minimize z1 + … + zm subject to bi - ai'x  zi , i = 1, … , m -bi + ai'x  zi , i = 1, … , m Quadratic error function can't be modeled as LP, but need calculus method (closed form solution)

  38. Special case of piecewise linear objective function : separable piecewise linear objective function. function f: Rn R, is called separable if f(x) = f1(x1) + f2(x2) + … + fn(xn) c1 < c2 < c3 < c4 fi(xi) c4 c3 slope: ci c2 c1 xi a1 a2 a3 0 x1i x4i x3i x2i

  39. Express xi in the constraints as xi x1i + x2i + x3i + x4i , where 0  x1i  a1, 0  x2i  a2 - a1 , 0  x3i  a3 - a2, 0  x4i In the objective function, use : min c1x1i + c2x2i + c3x3i + c4x4i Since we solve min problem, it is guaranteed that we get xki > 0 in an optimal solution implies xji , j < k have values at their upper bounds.

  40. 1.4 Graphical representation and solution • Let a  Rn, b R. Geometric intuition for the solution sets of { x : a’x = 0 } { x : a’x  0 } { x : a’x  0 } { x : a’x = b } { x : a’x  b } { x : a’x  b }

  41. { x : a’x  0 } { x : a’x = 0 } { x : a’x  0 } • Geometry in 2-D a 0

  42. Let z be a (any) point satisfying a’x = b. Then { x : a’x = b } = { x : a’x = a’z } = { x : a’(x – z) = 0 } Hence x – z = y, where y is any solution to a’y = 0, or x = y + z. Similarly, for { x : a’x  b }, { x : a’x  b }. { x : a’x  b } a z 0 { x : a’x  b } { x : a’x = b } { x : a’x = 0 }

  43. min c1x1 + c2x2 s.t. -x1 + x2 1, x1  0, x2  0 x2 c=(1, 0) c=(-1, -1) c=(1, 1) c=(0, 1) x1 {x: x1 + x2 = z} {x: x1 + x2 = 0}

  44. Representing complex solution set in 2-D ( n variables, m equations (coefficient vectors are linearly independent), nonnegativity, and n – m = 2 ) x3 x1 = 0 x2 x2 = 0 x3 = 0 x1 • See text sec. 1.5, 1.6 for more backgrounds

More Related