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Transcript
Linear Programming Piyush Kumar Welcome to COT 5405 Optimization For example Min cT x subject to Ax b x0 This is what is known as a standard linear program. Linear Programming Significance A lot of problems can be converted to LP formulation o Perceptrons (learning), Shortest path, max flow, MST, matching, … Accounts for major proportion of all scientific computations Helps in finding quick and dirty solutions to NP-hard optimization problems Both optimal (B&B) and approximate (rounding) Graphing 2-Dimensional LPs Optimal Solution y Example 1: 4 Maximize x+y Subject to: x + 2 y 2 x3 3 Feasible Region 2 y4 x0 y0 1 0 These LP animations were created by Keely Crowston. x 0 1 2 3 Graphing 2-Dimensional LPs y Example 2: 4 Minimize ** Multiple Optimal Solutions! x-y Subject to: 1/3 x + y 4 -2 x + 2 y 4 3 2 Feasible Region x3 x0 y0 1 0 0 1 2 3 x Graphing 2-Dimensional LPs y Example 3: 40 Minimize x + 1/3 y Subject to: x + y 20 -2 x + 5 y 150 30 Feasible Region 20 x5 x 0 y 0 10 x Optimal Solution 0 0 10 20 30 40 Do We Notice Anything From These 3 Examples? Extreme point y y y 4 4 40 3 3 30 2 2 20 1 1 10 0 0 1 2 3 x 0 0 1 2 3 x 0 x 0 10 20 30 40 A Fundamental Point y y y 4 4 40 3 3 30 2 2 20 1 1 10 0 0 1 2 3 x 0 0 1 2 3 x 0 x 0 10 20 30 If an optimal solution exists, there is always a corner point optimal solution! 40 Graphing 2-Dimensional LPs Second Corner pt. Example 1: Optimal Solution y 4 Maximize x+y Subject to: x + 2 y 2 x3 3 Feasible Region 2 y4 x 0 y 0 1 Initial 0 Corner pt. x 0 1 2 3 And We Can Extend this to Higher Dimensions Then How Might We Solve an LP? The constraints of an LP give rise to a geometrical shape - we call it a polyhedron. If we can determine all the corner points of the polyhedron, then we can calculate the objective value at these points and take the best one as our optimal solution. The Simplex Method intelligently moves from corner to corner until it can prove that it has found the optimal solution. But an Integer Program is Different y Feasible region is a set of discrete points. Can’t be assured a 4 3 corner point solution. There are no “efficient” 2 ways to solve an IP. Solving it as an LP provides a relaxation and a bound on the solution. 1 0 0 1 2 3 x Linear Programs in higher dimensions minimize subject to z= 7x1 + x2 + 5x3 x1 - x2 + 3x3 >= 10 5x1 + 2x2 - x1, x2, x3 x3 >= 6 0 What happens at (2,1,3)? What does it tell us about z* = optimal value of z? LP Upper bounds Any feasible solution to LP gives an upper bound on z* So now we know z* <= 30. How do we construct a lower bound? z* >= 16? [Y/N]? Lower bounding an LP 7x1+x2+5x3 >= (x1-x2+3x3) + (5x1+2x2-x3) >= 16 Find suitable multipliers ( >0 ?) to construct lower bounds. How do we choose the multipliers? The Dual maximize subject to z’ = 10y1 + 6y2 y1 + 5y2 <= 7 -y1 + 2y2 <= 1 3y1 – y1, y2 y2 <= 5 0 What is the dual of a dual? Every feasible solution of the dual gives a lower bound on z* The Primal minimize z= subject to 7x1 + x2 + 5x3 x1 - x2 + 3x3 >= 10 5x1 + 2x2 - x1, x2, x3 x3 >= 6 0 Every feasible solution of the primal is an upper bound on the solution to the dual. Primal – Dual picture Strong Optimality Primal = Dual at opt Z* 0 Dual Solutions Primal Solutions Duality A variable in the dual is paired with a constraint in the primal Objective function of the dual is determined by the right hand side of the primal constraints The constraint matrix of the dual is the transpose of the constraint matrix in the primal. Duality Properties Some relationships between the primal and dual problems: 1. If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable 2. If the optimal value of the primal is unbounded then the dual is infeasible. 3. If the optimal value of the dual is unbounded then the primal is infeasible. In Matrix terms Min cT x subject to Ax b x0 Amxn , cnx1 , xnx1 LP Geometry Forms a n dimensional polyhedron Is convex : If z1 and z2 are two feasible solutions then λz1+ (1- λ)z2 is also feasible. Extreme points can not be written as a convex combination of two feasible points. LP Geometry The normals to the halfspaces defining the polyhedron are formed by the coefficents of the constraints. Rows of A form the normals to the hyperplanes defining the primal LP pointing inside the polyhedron. LP Geometry Extreme point theorem: If there exists an optimal solution to an LP Problem, then there exists one extreme point where the optimum is achieved. Local optimum = Global Optimum LP: Algorithms Simplex. (Dantzig 1947) Developed shortly after WWII in response to logistical problems: used for 1948 Berlin airlift. Practical solution method that moves from one extreme point to a neighboring extreme point. Finite (exponential) complexity, but no polynomial implementation known. Courtesy Kevin Wayne LP: Polynomial Algorithms Ellipsoid. (Khachian 1979, 1980) Solvable in polynomial time: O(n4 L) bit operations. o n = # variables o L = # bits in input Theoretical tour de force. Not remotely practical. Karmarkar's algorithm. (Karmarkar 1984) O(n3.5 L). Polynomial and reasonably efficient implementations possible. Interior point algorithms. O(n3 L). Competitive with simplex! o Dominates on simplex for large problems. Extends to even more general problems. Ellipsoid Method Courtesy S. Boyd Barrier Algorithms Simplex solution path Barrier central path o Predictor o Corrector Optimum Interior Point Methods Back to LP Basics Standard form of LP Min cT x subject to Ax b x0 Amxn , cnx1 , xnx1 Standard form of the Dual Max bT y subject to AT y c y0 Amxn , cnx1 , xnx1 Weak Duality b y ( Ax ) y x A y x c c x T T T T T T We will not prove strong duality in this class but assume it. Complementary solutions For any primal feasible (but suboptimal) x, its complementary solution y is dual infeasible, with cx=yb For any primal optimal x*, its complementary solution y* is dual optimal, with cx*=y*b=z* Duality Gap = cx-yb Complementary slackness x*, y* are feasible, then they are optimal for (P) and (D) iff For I = 1..m if yi* > 0 o Then aix* = bi For J = 1..n if xj* > 0 o Then y*Aj = ci ai are rows of A and Aj are the columns of A Complementary slackness x*, y* are simultaneously optimal for (P) and (D) iff y*(Ax* - b) = 0 (y*A – c)x* = 0 Summary: If a variable is positive, its dual constraint is tight Or if a constraint is loose its dual variable is zero. Complementary Slackness Proof? y*(Ax* - b) - (y*A – c)x* = y*Ax* - y*b - y*Ax* + cx* = cx* - y*b =0 ( But all terms are non-negative ) Hence all must be zero! Primal-Dual Algorithms Find a feasible solution for both P and D. Try to satisfy the complementary slackness conditions. Algorithm Design Techniques LP Relaxation Rounding o Round the fractional solution obtained by solving LP-relaxation. o Runs fast Primal Dual Schema o (iteratively constructs primal n dual solutions) y objective LP optimum feasible solutions x Linear Program y objective optimum of LP relaxation IP optimum feasible solutions = rounding down optimum of LP relaxation x Integer Program Linear Relaxations What happens if the optimal of a LPRelaxation is Integral? There are a class of IPs for which this is guaranteed to happen Transportation problems MaxFlow problems In general (Unimodularity) … Exact Relaxation Lower Bounds Assume minimization problem Any relaxation of the original IP has a _____________ optimal objective function value than the optimal objective function value of the original IP z*relaxation z* z*relaxation is called a __________________ on z* Difference between these two values is called the relaxation gap Upper Bounds Any feasible solution to the original IP has a _____________ objective function value than the optimal objective function value of the original IP zfeasible z* zfeasible is called an __________________ on z* Heuristic techniques can be used to find “good” feasible solutions Efficient, may be beneficial if optimality can be sacrificed Usually application- or problem-specific Vertex Cover Introduction to LP Rounding A simple 2-approximation using LP Better than 2-factor approx?