OPTIMIZATION. joint course with. Ottimizzazione Discreta and Complementi di R.O. Edoardo Amaldi. DEIB Politecnico di Milano

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1 OPTIMIZATION joint course with Ottimizzazione Discreta and Complementi di R.O. Edoardo Amaldi DEIB Politecnico di Milano Website: Academic year Edoardo Amaldi (PoliMI) Optimization Academic year / 15

2 Chapter 1: Introduction Optimization is one of the most active branches of applied mathematics with a very wide range of relevant applications. General problem: Given a set X R n and a function f : X R to be minimized, find an optimal solution x X, i.e., such that f (x ) f (x) x X. Course s aim: Present the main concepts and methods of discrete and nonlinear (continuous) optimization, covering also modeling aspects. Prerequisites: linear programming, graph optimization (minimum spanning tree, shortest paths, maximum flow), and basics of NP-completeness theory. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

3 Many decision-making problems cannot be appropriately formulated or approximated in terms of linear models due to intrinsic nonlinearity. Examples 1) Production planning Determine the production levels so as to maximize the total profit while respecting the constraints on the availability of the various resources. - Since prices are elastic (amount that can be sold is inversely proportional to selling price) unit profit of a good decreases when the amount produced increases. - Due to economy of scale, the unit cost often decreases when the amount produced increases. 2) Discrete decisions modelled with binary/integer variables. Very special type of nonlinearity: x Z can be expressed as sin(πx) = 0. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

4 Examples of optimization problems and models 1) Location and transportation Given m warehouses, indexed with i = 1... m, with capacity p i and an area A i R 2 in which it can be located, n clients with known coordinates (a j, b j ) and demand d j, where j = 1... n, decide where to locate the warehouses and how to serve the clients so as to minimize the transportation costs (proportional to distance and amount of product) while respecting warehouse capacities and client demands. Assumptions: single type of product and m i=1 p i n j=1 d j Edoardo Amaldi (PoliMI) Optimization Academic year / 15

5 Decision variables: (x i, y i ) coordinates of the i-th warehouse, 1 i m w ij amount of product transported from warehouse i to client j, 1 i m and 1 j n distance between warehouse i and client j, 1 i m and 1 j n t ij Optimization model: min s.t. m n i=1 j=1 w ijt ij n j=1 w ij p i m i=1 w ij d j t ij = (x i a j ) 2 + (y i b j ) 2 i, j (1) i j (x i, y i ) A i R 2 i w ij 0 t ij 0 i, j i, j N.B.: t ij are not necessary, use equations (1) to substitute t ij in the objective function. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

6 2) Image reconstruction (Computer Tomography) Part of the body V R 3 subdivided into n small cubes V j called voxels. Assumption: matter density is constant within each voxel (e.g., pixels). Problem: Given the measurements provided by m beams, reconstruct a 3-D image of V, that is, determine the density x j for each V j. i-the beam intersects a subset of voxels indexed by J i {1,..., n}. For every j J i, let a ij denote the path length of i-th beam within voxel V j. i-th beam attenuation depends on the total amount of matter on the way: j J i a ij x j. Let b i be the measurement of the i-th beam at the exit point. Given m beams with prescribed directions, we obtain the linear system: a ij x j = b i i = 1,..., m j J i x j 0 j = 1,..., n usually infeasible due to measurement errors, non uniformity of the V j s,... Edoardo Amaldi (PoliMI) Optimization Academic year / 15

7 Possibile formulation min m i=1 (b i j J i a ij x j ) 2 s.t. x j 0 j = 1,..., n. Since the number of voxels is usually larger than the number of beams (m < n), to avoid alternative optimal solutions we may minimize: m f (x) = (b i n a ij x j ) 2 + δ x j with δ > 0 j J i i=1 j=1 Various possible choices for the objective function f (x), which may involve nonlinear terms accounting for the properties of matter/image stochastic model of attenuation and appropriate maximum likelihood estimator. Also the number of beams and their directions should be optimized. Current development: 4-D optimization (time) to account for respiratory motion. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

8 3) Combinatorial auctions With internet explosion the popularity of auctions has been growing. Participants (bidders) can place bids on combinations of discrete items, rather than individual items or continuous quantities. Examples: rare stamps or coins, airport time slots, wireless bandwidth, delivery routes, railroad segments,... Given a set of bidders N, a discrete set M of m distinct items being auctioned, for every subset of items S M, let b j (S) be the bid that bidder j N is willing to pay for subset S. Natural assumption: if S T = then b j (S) + b j (T ) b j (S T ) The bidder is willing to pay more for S T than for sets S and T individually (e.g., complete collection of rare stamps). Edoardo Amaldi (PoliMI) Optimization Academic year / 15

9 Key problem: Determine the winner of each item so as to maximize the total revenue. Let b(s) = max j N b j (S). For each S M, binary variable x S with x S = 1 if the highest bid on subset S is accepted, and x S = 0 otherwise. Problem formulation: max s.t. S M b(s)x S S M : i S x S 1 i M x S {0, 1} S M, involving 2 M variables, growing exponentially with the number of items. N.B.: when x S = 1, the subset of items S is given to a bidder willing to pay the largest amount. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

10 General optimization problem min f (x) s.t. g i (x) 0 1 i m x S R n - the algebraic and set constraints define the feasible region X = S {x R n : g i (x) 0, 1 i m}, - the objective function f (x) must be defined at least on X, namely f : X R, - the constraint functions g i (x) must be defined at least on S, namely g i : S R for i = 1,..., m. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

11 We just need to consider minimization problems since max{f (x) : x X } = min{ f (x) : x X }. Without loss of generality, we can also assume that all algebraic constraints are inequality constraints since g(x) = 0 { g(x) 0 g(x) 0. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

12 Solving a minimization problem to optimality amounts to finding a feasible solution that is globally optimal. Definition i) A feasible solution x X is a global optimum if f (x ) f (x) x X. ii) A feasible solution x X is a local optimum if ɛ > 0 such that f (x) f (x) x X N ɛ (x) where N ɛ (x) = {x X : x x ɛ}. For difficult problems, we have to settle for finding a good local optimum within a reasonable computing time. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

13 Main classes of optimization problems Terminology: programming optimization f g i S problem type linear linear S = R n Linear Programming (LP) linear linear S Z n Integer L. P. (ILP) linear linear S Z n 1 R n 2 with n = n 1 + n 2 Mixed Integer L. P. (MILP) at least one nonlinear S R n Nonlinear Programming (NLP) at least one nonlinear S Z n 1 R n 2 with n = n 1 + n 2 Mixed Integer NLP (MINLP) Some important special cases: Quadratic programming: f (x) = x T Qx + c T x with linear constraints Convex programming: f, the g i functions and S are, respectively, convex functions and convex set. Edoardo Amaldi (PoliMI) Optimization Academic year / 15

14 Some fields of application computational biology (determine the 3-D structure of proteins,...) health care planning and management (treatment planning, workforce scheduling, operating theater scheduling,...) optimal control (determine the trajectory of a robot arm, airplane, shuttle) logistics (location of plants and services, transportation, routing) and supply chain design and management data mining/machine learning: classification, clustering, approximation,... economics (risk management, portfolio optimization, combinatorial auctions, equilibria of games,... ) production planning and inventory management (manufacturing, chemical processes, energy generation,...) network planning and management (wired and wireless telecommunications, electric networks,...) Edoardo Amaldi (PoliMI) Optimization Academic year / 15

15 Some fields of application management of environmental and territorial resources (water, forest,...) design of experiments (for chemical and pharmaceutical companies) signal and image processing (2-D and 3-D reconstruction) statistics (e.g., nonlinear regression, estimation of distribution parameters) agriculture and agri-food industry dimensioning and optimization of structures (bridge, aircraft profile,...)... Edoardo Amaldi (PoliMI) Optimization Academic year / 15