Algorithms & Data Structures 2

Transcription

1 Algorithms & Data Structures 2 Optimization WS2017 B. Anzengruber-Tanase (Institute for Pervasive Computing, JKU Linz) (Institute for Pervasive Computing, JKU Linz)

2 WHAT IS OPTIMIZATION? Optimization problems are made up of three basic ingredients: An objective function which we want to minimize or maximize Examples: maximize the profit or minimize the cost in a manufacturing process in fitting experimental data to a user-defined model, minimize the total deviation of observed data from predictions based on the model In designing an automobile panel, we might want to maximize the strength. A set of unknowns or variables which affect the value of the objective function Examples: In the manufacturing problem, the variables might include the amounts of different resources used or the time spent on each activity In fitting-the-data problem, the unknowns are the parameters that define the model In the panel design problem, the variables used define the shape and dimensions of the panel. A set of constraints that allow the unknowns to take on certain values but exclude others Examples: For the manufacturing problem, it does not make sense to spend a negative amount of time on any activity, so we constrain all the "time" variables to be non-negative. In the panel design problem, we would probably want to limit the weight of the product and to constrain its shape. The optimization problem is then: Find values of the variables that minimize or maximize the objective function while satisfying the constraints. Algorithms & Datastructures 2 // 2017W // 2

3 Optimization CLASSIFICATION The word "Programming" is used here in the sense of "planning" Feasability Problem Discrete Integer Programming Stochastic Programming Linear Programming Semidefinite Progr. Nonlinearly Constrained Single Objective Function Bound Constrained Quadratic Programming Network Programming Multiobjective Optimization Continuous Constrained Stochastic Programming Nonlinear Equations Unconstrained Nonlinear Least Squares Global Optimization Nondifferentiable Opt. Algorithms & Datastructures 2 // 2017W // 3

4 Optimization CLASSIFICATION The word "Programming" is used here in the sense of "planning" Feasability Problem Discrete Integer Programming Stochastic Programming Linear Programming Semidefinite Progr. Nonlinearly Constrained Single Objective Function Bound Constrained Quadratic Programming Network Programming Multiobjective Optimization Continuous Constrained Stochastic Programming Nonlinear Equations Unconstrained Nonlinear Least Squares Global Optimization Nondifferentiable Opt. Algorithms & Datastructures 2 // 2017W // 4

5 FEASIBILITY PROBLEM In some cases the goal is to find a set of variables that satisfies the constraints of the model. The user does not particularly want to optimize anything so there is no reason to define an objective function. This type of problems is usually called a feasibility problem or constraint satisfaction problem. Examples: 8 Queens Problem (n-queens Problem) Graph Colouring Solution Techniques: From the standpoint of computational complexity, finding out if problem has a feasible solution might be essentially as hard as actually finding the optimal solution Because if there exists no feasible solution, then the entire solution space must be explored to prove this. There are no shortcuts in general, unless something useful about the model's structure is known (e.g., if solving some form of a transportation problem, it may be possible to assure feasibility by checking that the sources add up to at least as great a number as the sum of the destinations). Algorithms & Datastructures 2 // 2017W // 5

6 Optimization CLASSIFICATION The word "Programming" is used here in the sense of "planning" Feasability Problem Discrete Integer Programming Stochastic Programming Linear Programming Semidefinite Progr. Nonlinearly Constrained Single Objective Function Bound Constrained Quadratic Programming Network Programming Multiobjective Optimization Continuous Constrained Stochastic Programming Nonlinear Equations Unconstrained Nonlinear Least Squares Global Optimization Nondifferentiable Opt. Algorithms & Datastructures 2 // 2017W // 6

7 MULTIPLE OBJECTIVE FUNCTIONS Optimization of a number of different objectives at once. Usually, the different objectives are not compatible, i.e. the variables that optimize one objective may be far from optimal for the others. Example: In the panel design problem, it would be nice to minimize weight and maximize strength simultaneously. Solution Techniques Goal Programming: problems with multiple objectives are reformulated as single-objective problems by either forming a weighted combination of the different objectives or by replacing some of the objectives by constraints Pareto preference analysis essentially brute force examination of all possible solutions Priority Ordering Put objective functions in priority order, optimize on one objective, then change it to a constraint fixed at the optimal value (perhaps subject to a small tolerance), and repeat with the next function. Algorithms & Datastructures 2 // 2017W // 7

8 Optimization CLASSIFICATION The word "Programming" is used here in the sense of "planning" Feasability Problem Discrete Integer Programming Stochastic Programming Linear Programming Semidefinite Progr. Nonlinearly Constrained Single Objective Function Bound Constrained Quadratic Programming Network Programming Multiobjective Optimization Continuous Constrained Stochastic Programming Nonlinear Equations Unconstrained Nonlinear Least Squares Global Optimization Nondifferentiable Opt. Algorithms & Datastructures 2 // 2017W // 8

9 Constraints linear any SINGLE OBJECTIVE FUNCTION Constrained Continuous Optimization all variables are allowed to take values from subintervals of real numbers the values that variables are allowed to take are bounded by constraints Solution Techniques depend on characteristics of objective function, constraints, and variables variables deterministic stochastic objective function any quadratic linear Nonlinearly Constrained Algorithms & Datastructures 2 // 2017W // 9 Network Programming Bound Constrained Quadratic Programming Linear Programming Semidefinite Progr. Stochastic Programming

10 LINEAR PROGRAMMING (LP) The basic problem of linear programming is to minimize a linear objective function of continuous real variables, subject to linear constraints. For purposes of describing and analyzing algorithms, the problem is often stated in the standard form min{c T x: Ax = b, x 0 } or min{c T x: Ax + y = b, L (x, y) U} where xr n is the vector of unknowns, cr n... cost vector, AR mxn... constraint matrix. allowing general bounds on the variables The feasible region described by the constraints is a polytope, or simplex, and at least one member of the solution set lies at a vertex (corner) of this polytope. Example: min 3x 1-2x 2 s.t. x 1, x 2 0 x 1 19 x x 1 + x 2 30 x 1 - x 2 5 Algorithms & Datastructures 2 // 2017W // 10 brute force solution: compute value of objective function for each corner find minimum x 1 x 2 c T x /3 20/3 65/3 3/ x 2 x x 1-2x 2 =c geometric solution: draw line 3x 1-2x 2 =c reduce c, until 3x 1-2x 2 =c leaves feasible region x x 1 2x 1 + x 2 30 x 1 - x 2 5

11 LINEAR PROGRAMMING (LP) Solution Techniques: A is generally not square, hence it is not possible to solve an LP by just inverting A. Usually A has more columns (unknowns) than rows (constraints) (n>m), and Ax=b is therefore quite likely to be under-determined, leaving great latitude in the choice of x with which to minimize c T x. Brute-Force possible, but there exist better techniques: The simplex algorithm (so named because of the geometry of the feasible set) underlies the vast majority of available software packages for linear programming generates a sequence of feasible iterates by repeatedly moving from one vertex (corner) of the feasible set to an adjacent vertex with a lower value of the objective function. When it is not possible to find an adjoining vertex with a lower value of the objective function, the current vertex must be optimal, and termination occurs. Note, that computational complexity in the worst case is equal to brute force (all corners are visited) Barrier or interior-point methods, by contrast, visit points within the interior of the feasible region. These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar's innovative analysis in Most interior-point methods have polynomial complexity. Algorithms & Datastructures 2 // 2017W // 11

12 SIMPLEX METHOD Algebraically speaking, the simplex method is based on observation that at least n-m of the components of x are zero if x is a vertex of the feasible set. Accordingly, the components of x can be partitioned at each vertex into a set of m basic variables - all nonnegative - and a set of n-m nonbasic variables-all zero. If we gather the basic variables into a subvector, x B R m, and the nonbasics into another subvector, x N R n-m, we can partition the columns of A as [B N], where B contains the m columns that correspond to x B. (B is then square.) At each iteration of the simplex method, a basic variable (a component of x B ) is reclassified as nonbasic, and vice versa, i.e. x B and x N exchange a component. (Geometrically, this swapping process corresponds to a move from one vertex of the feasible set to an adjacent vertex.) We therefore need to choose which component of x N should enter x B (that is, be allowed to move off its zero bound) and which component of x B should enter x N (that is, be driven to zero). In fact, we need make only the first of these choices, since the second choice is implied by the feasibility constraints Ax=b and x 0. Algorithms & Datastructures 2 // 2017W // 12

13 SIMPLEX METHOD (CONTD.) In selecting the entering component, we note that c T x can be expressed as a function of x N alone. We can express x B in terms of x N by noting that Ax=b implies that x B = B -1 (b - N x N ). Hence, partitioning c into c B and c N in the obvious way, we have c T x = c BT x B + c NT x N = c NT B -1 b + d nt x n where the vector d n = [c N -N T B -T c B ] is called the reduced-cost vector. If all components of x B are strictly positive and some component (say, the i th component) of d n is negative, we can decrease the value of c T x by allowing component i of x N to become positive while adjusting x B to maintain feasibility. Unless there exist feasible points x that make c T x arbitrarily negative, the requirement x B 0 imposes an upper bound on x N,i, the i th component of x N. In principle, can choose any component x N,i with d N,i <0 as entering variable. If there are no negative entries in d n, the current point x is optimal. If there is more than one, we would ideally pick the component that will lead to the largest reduction in c T x on the current iteration: x N,i = min{(b -1 -b) j /(B -1 N i ) j : (B -1 N i ) j > 0} The index j that achieves the minimum in this formula indicates the basic variable x j that is to become nonbasic. If more than one such component achieves the minimum simultaneously, the one with the largest value of (B -1 N i ) is usually selected. Algorithms & Datastructures 2 // 2017W // 13

14 If all elements in pivot column are negative, then problem does not have a feasible solution -> stop! SIMPLEX METHOD USING A TABLEAU Simplex Iteration: If all d n >0, the x B is optimal solution Else choose column c with minimum value of d n ( pivot column, with x N,m+c being the variable entering the set of basic variables) choose row r with minimum value of b i /a ij considering only those rows where a ij >0 ( pivot row, with x B,r leaving the basis) use a rc as the pivot element Compute new tableau values as follows Exchange x B,r and x n,m+c Compute new coefficient values a rj in pivot row:and new basic variable value Compute new coefficient values in pivot column:and new reduced cost value Compute all other new coefficient values and new basic variable values and new reduced cost values Compute new value of objective function: Algorithms & Datastructures 2 // 2017W // 14 Simplex Tableau: basic variables non-basic variables name value x N,m+1 x N,m+2... x N,n x B, 1 b 1 x B, 2 b coefficients a ij.. with i=1..m and j=1..(n-m) x B, m b m value of d N,m+1 d N,m+2... d N,n c T x c (reduced costs) 1 / a rc for j = c a rj = a rj / a rc else b r = b r / a rc a ic = a ic / a rc for i r d c = d c / a rc a ij = a ij a rj * a ic for i r, j c b i = b i b r * a ic for i r d j = d j d c * a rj for j c c = c + b r * d c

15 SIMPLEX METHOD EXAMPLE Consider the following LP minimize 4x 1 6x 2 + 9x s.t. x i 0 for i=1..8 2x 1 x 2 + 3x 3 = 1 x 4 3x 3 = 2 x 5 2x 1 = 6 x 6 x 2 = 6 x 7-2x 1 + x 2 3x 3 = 2 x 8 exchange variables x 1 x 2 x 3 x x x x x Note: problem is given in so called canonical form, which can be translated directly into tableau. If problem is given in another form, then transformations are necessary (including for example the introduction of additional variables) Selecting Pivot row: min b i /a i2 for all i where a i2 >0 i.e. min{6/1;2/1} i.e. choose x 8 Algorithms & Datastructures 2 // 2017W // 15 pivot element Pivot column, defined by minimum negative reduced cost value

16 SIMPLEX METHOD EXAMPLE Consider the following LP minimize 4x 1 6x 2 + 9x s.t. x i 0 for i=1..8 2x 1 x 2 + 3x 3 = 1 x 4 3x 3 = 2 x 5 2x 1 = 6 x 6 x 2 = 6 x 7-2x 1 + x 2 3x 3 = 2 x 8 previous tableau for reference x 1 x 8 x 3 x x x x x x 1 x 2 x 3 x x x x x Compute new value of pivot element a rc = 1/ a rc = 1/1 = 1 Compute pivot row with a rj = a rj / a rc for j c and b r = b r / a rc (values not changed, because pivot element was 1) Algorithms & Datastructures 2 // 2017W // 16

17 SIMPLEX METHOD EXAMPLE Consider the following LP minimize 4x 1 6x 2 + 9x s.t. x i 0 for i=1..8 2x 1 x 2 + 3x 3 = 1 x 4 3x 3 = 2 x 5 2x 1 = 6 x 6 x 2 = 6 x 7-2x 1 + x 2 3x 3 = 2 x 8 Algorithms & Datastructures 2 // 2017W // 17 Compute new basic variable values b i = b i b r a ic for i r b 1 = b 1 -b 5 a 12 =1-2(-1) = 3 b 2 = b 2 -b 5 a 22 =2-20 = 2 b 3 = b 3 -b 5 a 32 =6-20 = 6 b 4 = b 4 -b 5 a 42 =6-21 = 4 Compute new value of objective function: c = c + b r d c c = c + b 5 d 2 = (-6) = 54 Compute pivot column with a ic = a ic / a rc for i r and d c = d c / a rc previous tableau for reference x 1 x 8 x 3 x x x x x Compute all other new coefficient values: a ij = a ij a rj a ic for i r, j c a 11 = a 11 -a 51 a 12 =2-(-2)(-1) = 0 a 21 = a 21 -a 51 a 22 =0-(-2)0 = 0 a 31 = a 31 -a 51 a 32 =2-(-2)0 = 2 a 41 = a 41 -a 51 a 42 =0-(-2)1 = 2 a 13 = a 13 -a 53 a 12 =3-(-3)(-1) = 0 a 23 = a 23 -a 53 a 22 =3-(-3)0 = 3 a 33 = a 33 -a 53 a 32 =0-(-3)0 = 0 a 43 = a 43 -a 53 a 42 =0-(-3)1 = 3 Compute new reduced cost values d j = d j d c a rj for j c x 1 x 2 x 3 x x x x x d 1 = d 1 d c a 51 = 4-(-6)(-2)= -8 d 3 = d 3 d c a 53 = 9-(-6)(-3)=

18 SIMPLEX METHOD EXAMPLE Algorithms & Datastructures 2 // 2017W // 18 x 1 x 8 x 3 x x x x x Optimum solution obtained! x 4 =3 x 5 =2 x 6 =2 x 1 =2 x 2 =6 x 7 =x 8 =x 3 =0 x 1 x 8 x 5 x x 3 2/ /3 x x x x 7 x 8 x 3 x x x x 1 2 1/2-1/2 3/2 x x 7 x 8 x 5 x x 3 2/ /3 x x 1 1 1/2 1-1/2 x

19 SEMIDEFINITE PROGRAMMING (SDP) The (linear) semidefinite programming problem (SDP) is essentially an ordinary linear program where the nonnegativity constraint is replaced by a semidefinite constraint on matrix variables. The standard form for the problem is min{c X: A k X = b k (k=1..m); X 0} where C, A k and X are all symmetric (nxn) matrices, b k is a scalar, and the constraint X 0 means that X, the unknown matrix, must lie in the closed, convex cone of positive semidefinite. Here, refers to the standard inner product on the space of symmetric matrices. Solution Technique Interior-Point Algorithms Algorithms & Datastructures 2 // 2017W // 19

20 NONLINEARLY CONSTRAINED PROGRAMMING (NLP) The general constrained optimization problem is to minimize a nonlinear function subject to nonlinear constraints. Two equivalent formulations of this problem are useful for describing algorithms. They are min{f(x): c i (x) 0, i I, c i (x) = 0, i E} (1) where each ci is a mapping from R n to R n, and I and E are index sets for inequality and equality constraints, respectively; and min{f(x): c(x) = 0, l x u} (2) where c maps from R n to R n, and the lower- and upper-bound vectors, l and u, may contain some infinite components. Solution Techniques The main techniques that have been proposed for solving constrained optimization problems are reduced-gradient methods, sequential linear and quadratic programming methods, and methods based on augmented Lagrangians and exact penalty functions. Algorithms & Datastructures 2 // 2017W // 20

21 BOUND CONSTRAINED OPTIMIZATION Bound-constrained optimization problems play an important role in the development of software for the general constrained problem because many constrained codes reduce the solution of the general problem to the solution of a sequence of bound-constrained problems. The problem is formulated as follows min{f(x): l x u} where the lower- and upper-bound vectors, l and u, may contain some infinite components Solution Techniques Newton Methods Gradient-Projection Methods Algorithms & Datastructures 2 // 2017W // 21

22 QUADRATIC PROGRAMMING (QP) The quadratic programming problem involves minimization of a quadratic function subject to linear constraints. The following formulation is commonly used: min{½ x T Qx+c T x: c it x b i, i I, c i Tx = b i, i E} where Q R n x R n respectively. is symmetric, and the index sets I and E specify the inequality and equality constraints, Solution Techniques The difficulty of solving the quadratic programming problem depends largely on the nature of the matrix Q. In convex quadratic programs, which are relatively easy to solve, the matrix Q is positive semidefinite. If Q has negative eigenvalues - nonconvex quadratic programming - then the objective function may have more than one local minimizer. Algorithms & Datastructures 2 // 2017W // 22

23 NETWORK PROGRAMMING Network problems arise in applications that can be represented as the flow of a commodity in a network. The resulting programs can be linear or non-linear. Due to the network structure of the model, fast techniques for solving these problems can be developed. Examples: Transportation Problem. We have a commodity that can be produced in different locations and needs to be shipped to different distribution centers. Given the cost of shipping a unit of commodity between each two points, the capacity of each production center, and the demand at each distribution center, find the minimal cost shipping plan. Assignment Problem (special case of the transportation problem). There are n individuals that need to be assigned to n different tasks. Each individual is assigned to one job only and each job is performed by one person. Given the cost that each individual charges for performing each of the n jobs, find a minimal cost assignment. Maximum Value Flow. Given a directed network of roads that connects two cities and the capacities of these roads, find the maximum number of units (cars) that can be routed from one city to another. Here, the constraints are the equilibrium or balance equations at each node (or road intersection); i.e., flow of the cars into a node is equal to the flow of the cars out of that node. Shortest Path Problem. Given a directed network and the length of each arc in this network, find a shortest between two given nodes. Minimum Cost Flow Problem. Given a directed network with upper and lower capacities on each of its arcs, and given a set of external flows (positive or negative) that need to be routed through this network, find the minimal cost routing of the given flows through this network. Here, the cost per unit of flow on each arc is assumed to be known. Algorithms & Datastructures 2 // 2017W // 23

24 STOCHASTIC PROGRAMMING All of the model formulations encountered thus far in the Optimization Tree have assumed that the data for the given problem are known accurately. However, for many actual problems, the problem data cannot be known accurately for a variety of reasons, e.g. due to simple measurement error that some data represent information about the future (e.g., product demand or price for a future time period) and simply cannot be known with certainty Problem Formulation It is written in the form of a mathematical program extended to a parameter space whose values are random variables (generally with known distribution function). Solution Techniques: forming a certainty equivalent. Average value. Replace all random variables with their means. Chance constraint. Given a stochastic program with a random variable, p, in its constraint: g(x; p) <= 0, a certainty equivalent is to replace this with the constraint, P[g(x; p) <= 0] >= a, where P[] is a (known) probability function on the range of g, and a is some acceptance level (a=1 means the constraint must hold for all values of p, except on a set of measure zero) Recourse model (see integer programming) Algorithms & Datastructures 2 // 2017W // 24

25 Optimization CLASSIFICATION The word "Programming" is used here in the sense of "planning" Feasability Problem Discrete Integer Programming Stochastic Programming Linear Programming Semidefinite Progr. Nonlinearly Constrained Single Objective Function Bound Constrained Quadratic Programming Network Programming Multiobjective Optimization Continuous Constrained Stochastic Programming Nonlinear Equations Unconstrained Nonlinear Least Squares Global Optimization Nondifferentiable Opt. Algorithms & Datastructures 2 // 2017W // 25

26 SINGLE OBJECTIVE FUNCTION Unconstrained Continuous Optimization In the unconstrained optimization problem, we seek a local minimizer of a real-valued function, f(x), where x is a vector of real variables. In other words, we seek a vector, x*, such that f(x*) <= f(x) for all x close to x*. Note: It's been argued that almost all problems really do have constraints. For example, any variable denoting the "number of objects" in a system can only be useful if it is less than the number of elementary particles in the known universe! In practice though, answers that make good sense in terms of the underlying physical or economic problem can often be obtained without putting constraints on the variables. Algorithms & Datastructures 2 // 2017W // 26

27 NONLINEAR LEAST SQUARES PROBLEM Least squares problems often arise in data-fitting applications. The nonlinear least squares problem has the general form min{r(x): x R n } where r is the function defined by r(x)= f(x) 2 for some vector-valued function f that maps R n to R n. Solution Techniques Gauss-Newton Method Levenberg-Marquardt Method Algorithms & Datastructures 2 // 2017W // 27

28 NONLINEAR EQUATIONS Systems of nonlinear equations arise as constraints in optimization problems, but also arise, for example, when differential and integral equations are discretized. In solving a system of nonlinear equations, we seek a vector such that f(x)=0 where x is an n-dimensional vector of n variables. Most solution algorithms are closely related to algorithms for unconstrained optimization and nonlinear least squares. Indeed, algorithms for systems of nonlinear equations usually proceed by seeking a local minimizer to the problem min{ f(x) : x R n } for some norm, usually the 2-norm. This strategy is reasonable, since any solution of the nonlinear equations is a global solution of the minimization problem. Solution Techniques Newton s Method Algorithms & Datastructures 2 // 2017W // 28

29 GLOBAL OPTIMIZATION Global optimization algorithms try to find an x* that minimizes f(x) over all possible vectors x. Solution Techniques any (purposeful) search procedure to seek a solution to a global optimization problem blind search methods include: Breadth-first: expanding a node of least depth Depth-first: expanding a node of greatest depth (requiring backtracking) Search strategies guided by a heuristic function Branch-and-Bound Heuristic search strategies are often based on some biological metaphor: Ant colony optimization, based on how ants solve problems Genetic algorithm, based on genetics and evolution Neural networks, based on how the brain functions Simulated annealing, based on thermodynamics Tabu search, based on memory-response Target analysis, based on learning Algorithms & Datastructures 2 // 2017W // 29

30 NONDIFFERENTIABLE OPTIMIZATION Problems that require the optimization of a nondifferentiable function Problems that require the optimization of a nondifferentiable function often occur in the efficient solution of large linear programming problems and in the context of Lagrangian duality. Solution Techniques Interior Point Method Cutting-Plane Method Algorithms & Datastructures 2 // 2017W // 30

31 Optimization CLASSIFICATION The word "Programming" is used here in the sense of "planning" Feasability Problem Discrete Integer Programming Stochastic Programming Linear Programming Semidefinite Progr. Nonlinearly Constrained Single Objective Function Bound Constrained Quadratic Programming Network Programming Multiobjective Optimization Continuous Constrained Stochastic Programming Nonlinear Equations Unconstrained Nonlinear Least Squares Global Optimization Nondifferentiable Opt. Algorithms & Datastructures 2 // 2017W // 31

32 SINGLE OBJECTIVE FUNCTION Discrete Optimization some ( mixed integer programming ) or all ( integer programming ) of the variables must be integer values Example Maximum flow in networks (with discrete edge capacities) Special Case: 0/1 integer programming problems e.g.: Knapsack Problem (variables are either 0 (item is not taken) or 1 (item is taken) Travelling Salesman Problem Stochastic Programming Recourse Problems are staged problems wherein one alternates decisions with realizations of stochastic data. The objective is to minimize total expected costs of all decisions. Algorithms & Datastructures 2 // 2017W // 32

33 INTEGER LINEAR PROGRAMMING In many applications, the solution of an optimization problem makes sense only if certain of the unknowns are integers. Integer linear programming problems have the general form min{c T x: Ax = b, x 0, x Z n } where x is the set of n-dimensional integer vectors. Solution Techniques It may not be obvious that integer programming is a very much harder problem than ordinary linear programming, but that is nonetheless the case, in both theory and practice. Branch and Bound Choose non-integer Solution x for Problem Build enumeration tree Left child: subproblem with x i = x i Right child: subproblem with x i = x i Traversal (DFS/BFS) and branching order depends on particular B&B strategy Depth of tree corresponds to dimension of x (number of integer variables), in a leave node all variables must be integer values Branches can be pruned by maintaining lower bounds on the value of the objective function, thus not the whole tree must be expanded (at least in the average case! In the worst case e.g. if a weak bounding function is used all leaves must be visited and the strategy is equal to brute force) Algorithms & Datastructures 2 // 2017W // 33

34 Simulated Annealing

35 CONTINUOUS (GLOBAL) OPTIMIZATION given a continuous cost function F find the minimum/maximum value of F across a defined domain of variables (= solution space) gradient method liability: get stuck in local optimum Algorithms & Datastructures 2 // 2017W // 36

36 ANNEALING avoid gradient method liability: controlled counter-stepping Use probabilistic control for counter-steps, i.e. adopt thermodynamics of the annealing process modeled using the Boltzmann probability distribution: Prob(E) ~ exp(-e/kt) E... possible energy state T... system temperature k... Boltzmann s constant Algorithms & Datastructures 2 // 2017W // 37

37 THE BOLTZMANN DISTRIBUTION Algorithms & Datastructures 2 // 2017W // 38

38 METROPOLIS ALGORITHM [METROPOLIS 53] originally: computer simulation of thermodynamic processes 1. An initial configuration, C i is chosen at random or by some other, usually heuristic method. This configuration has an associated cost, E i = F(C i ). 2. An initial temperature, T, is determined. 3. A random change in the configuration is generated. This change produces a possible new configuration, Ci+1. Energy = objective Function 4. If E i+1 E i, then change is always accepted. 5. If E i+1 > E i, then change is accepted with probability exp[-(e i+1 E i )k T] Probabilistic Step 6. Update temperature T if required. 7. Repeat from 2. until T < T stop cooling schedule stopping criterion Algorithms & Datastructures 2 // 2017W // 39

39 SIMULATED ANNEALING reasonable probability for up-hill stepping at high temperature moderate probability for up-hill stepping at low temperature Algorithms & Datastructures 2 // 2017W // 40

40 SIMULATED ANNEALING ALGORITHM Anneal(I,S) PRE: POST: I = Instance of problem S = initial feasible solution to I of size n > 0 h(s) = objective function, cost of solution S = feasible solution for I Determine initial temperature T repeat tries = moves = goodmoves := 0 repeat NewS := random neighbour of S S := h(news) - h(s) tries := tries + 1 rnd := random number between 0 and 1 if S < 0 or rnd < e -S/T then S := NewS moves := moves + 1 if S < 0 then goodmoves := goodmoves + 1 until goodmoves>n or moves>2*n or tries>20*n T := 0.85*T Until moves < 0.05 tries or T < e Algorithms & Datastructures 2 // 2017W // 41

41 SETTING AN INITIAL TEMPERATURE Temperature(T) POST: T provides about 95% chance of changing solutions select an initial solution S 0 Repeat n times generate a neighboring solution S compute S := h(s 0 ) - h(s) if S > 0 then sum := sum + S set ave = sum/n find a T such that 0.95 < e -ave/t Algorithms & Datastructures 2 // 2017W // 42

42 BRANCH AND BOUND Born in integer programming (IP), B&B is more generally used in global optimization, including continuous functions. Algorithm Sketch: 1. Initialize set list with feasible region (X plus all constraints), say S, with associated bounds -inf, inf (finite bounds on f can be computed, if computationally preferable, and the initial set could be a superset of the feasible region). Initialize f* equal to the lower bound (-inf, unless an initial feasible solution is known). 2. Branch by selecting the k-th set from list, say S_k, and splitting into a partition, say S_k = A\/B such that the interior of A does not intersect the interior of B. (Could partition S_k into more than 2 subsets.) 3. Bound f in A and/or in B, say L(S) <= f(x) <= U(S) for all x in S, for S = A and/or B. Define U(S) = -inf if it is determined that S is empty (in which case S will be discarded in the next step). 4. Test if U(S) <= f* + t, for S = A and/or B. If not, put S on list. (If the test passes, S is discarded by not being put on the list. Note: f* is best solution value to date, and t is a tolerance that says we want to get a solution whose objective value is within t of the optimal value.) Further, if L(S) > f* and the associated point, x(s), is feasible (where f(x(s)) = L(S)), replace f* with L(S) and x* with x(s). If the list is not empty, go branch. B&B terminates when the list becomes empty, and the best feasible solution obtained is x* with value f*, which is within t of optimality. (If f* = -inf, there is no feasible solution.) Algorithms & Datastructures 2 // 2017W // 43

43 BRANCH AND BOUND Notes A particular bounding rule for IP is to solve the LP relaxation, whose value is an upper bound on the integervalued max. The lower bounds do not change unless an integer solution is obtained from the LP, in which case the best solution to date can change. Branching rules can range from breadth-first: expand all possible branches from a node in the search tree before going deeper into the tree, to depth-first: expand the deepest node first (and backtrack, as necessary). A best-first search is to select the node (i.e., set on list) that seems most promising, such as one with the greatest upper bound. Typically, there are multiple criteria for branching, and these change depending on whether a feasible solution has been obtained. Backtracking need not go in exactly the reverse order (i.e., need not have a LIFO rule), but care must be taken when reordering the path. When the problem is not finite (not IP), branching rules need to consider the hyper-volume of each set in a partition to ensure convergence. Algorithms & Datastructures 2 // 2017W // 44

44 BRANCH AND BOUND EXAMPLE 5 4 x 2 Optimum solution (S): x 1 =2,2 x 2 =2,8 x 1 + 2x 2 = 7,73 Consider the following (M)ILP maximize x 1 + 2x 2 s.t. 6x 1 + 5x 2 27 (a) 2x 1 + 6x 2 21 (b) x 1, x 2 0 x 1, x (a) x 1 + 2x 2 (b) x 1 Algorithms & Datastructures 2 // 2017W // 45

45 BRANCH AND BOUND EXAMPLE x 2 Splitting into Partitions Solution S1 x 1 = 2 x 2 = 2,83 x 1 + 2x 2 = 7,67 S1 S Solution = 7,73 x 1 = 2,2 x 2 = 2,8 x 1 2 x 1 3 Solution = 7,67 x 1 = 2 x 2 = 2,83 S2 Solution = 6,6 x 1 = 3 x 2 = 1,8 2 1 S1 S2 (a) x 1 + 2x 2 (b) Solution S2 x 1 = 3 x 2 = 1,8 x 1 + 2x 2 = 6, x 1 Algorithms & Datastructures 2 // 2017W // 46

46 BRANCH AND BOUND EXAMPLE x 2 S4 Solution S4 x 1 = 1,5 x 2 = 3 x 1 + 2x 2 = 7,5 x 2 2 S1 S Solution = 7,73 x 1 = 2,2 x 2 = 2,8 x 1 2 x 1 3 Solution = 7,67 x 1 = 2 x 2 = 2,83 x 2 3 S2 Solution = 6,6 x 1 = 3 x 2 = 1,8 2 1 S3 (a) x 1 + 2x 2 (b) S3 Solution = 6 x 1 = 2 x 2 = 2 S4 Solution = 7,5 x 1 = 1,5 x 2 = x 1 Solution S3 x 1 = 2 x 2 = 2 x 1 + 2x 2 = 6 Algorithms & Datastructures 2 // 2017W // 47

47 BRANCH AND BOUND EXAMPLE x 2 S5 (a) Solution S5 x 1 = 1 x 2 = 3,17 x 1 + 2x 2 = 7,33 x 1 + 2x 2 (b) x 1 x 2 2 S3 S1 S Solution = 7,67 x 1 = 2 x 2 = 2,83 Solution = 7,73 x 1 = 2,2 x 2 = 2,8 x 1 2 x 1 3 Solution = 6 x 1 = 2 x 2 = 2 S5 x 2 3 x 1 1 S4 S2 Solution = 7,33 x 1 = 1 x 2 = 3,17 Solution = 6,6 x 1 = 3 x 2 = 1,8 Solution = 7,5 x 1 = 1,5 x 2 = 3 x 1 2 S6 Algorithms & Datastructures 2 // 2017W // 48

48 BRANCH AND BOUND EXAMPLE x 2 S7 x 2 2 S1 S Solution = 7,73 x 1 = 2,2 x 2 = 2,8 x 1 2 x 1 3 Solution = 7,67 x 1 = 2 x 2 = 2,83 x 2 3 S2 Solution = 6,6 x 1 = 3 x 2 = 1,8 2 1 (a) x 1 + 2x 2 (b) S3 Solution = 6 x 1 = 2 x 2 = 2 x 1 1 S4 Solution = 7,5 x 1 = 1,5 x 2 = 3 x Optimal Solution S7: x 1 = 1 x 2 = 3 x 1 + 2x 2 = 7 x 1 x 2 3 S5 Solution = 7,33 x 1 = 1 x 2 = 3,17 x 2 4 S6 Algorithms & Datastructures 2 // 2017W // 49 S7 Solution = 7 x 1 = 1 x 2 = 3 S8

49 STOCHASTIC PROGRAMMING (DISCRETE) Recourse The fundamental idea behind stochastic linear programming is the concept of recourse. Recourse is the ability to take corrective action after a random event has taken place. A simple example of two-stage recourse is the following: Choose some variables, x, to control what happens today. Overnight, a random event happens. Tomorrow, take some recourse action, y, to correct what may have gotten messed up by the random event. We can formulate optimization problems to choose x and y in an optimal way. In this example, there are two periods; the data for the first period are known with certainty and some data for the future periods are stochastic, that is, random. Stochastic programs seek to minimize the cost of the first-period decision plus the expected cost of the secondperiod recourse decision. This concept can be extended to recourse models with more than two stages. Algorithms & Datastructures 2 // 2017W // 50

50 Algorithms & Data Structures 2 Optimization WS2017 B. Anzengruber-Tanase (Institute for Pervasive Computing, JKU Linz) (Institute for Pervasive Computing, JKU Linz)