A Fast Algorithm for Maximum Likelihood Estimation of Mixture Proportions Using Sequential Quadratic Programming
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4. Convex Optimization Problems
Convex Optimization — Boyd & Vandenberghe 4. Convex optimization problems optimization problem in standard form • convex optimization problems • quasiconvex optimization • linear optimization • quadratic optimization • geometric programming • generalized inequality constraints • semidefinite programming • vector optimization • 4–1 Optimization problem in standard form minimize f0(x) subject to f (x) 0, i =1,...,m i ≤ hi(x)=0, i =1,...,p x Rn is the optimization variable • ∈ f : Rn R is the objective or cost function • 0 → f : Rn R, i =1,...,m, are the inequality constraint functions • i → h : Rn R are the equality constraint functions • i → optimal value: p⋆ = inf f (x) f (x) 0, i =1,...,m, h (x)=0, i =1,...,p { 0 | i ≤ i } p⋆ = if problem is infeasible (no x satisfies the constraints) • ∞ p⋆ = if problem is unbounded below • −∞ Convex optimization problems 4–2 Optimal and locally optimal points x is feasible if x dom f and it satisfies the constraints ∈ 0 ⋆ a feasible x is optimal if f0(x)= p ; Xopt is the set of optimal points x is locally optimal if there is an R> 0 such that x is optimal for minimize (over z) f0(z) subject to fi(z) 0, i =1,...,m, hi(z)=0, i =1,...,p z x≤ R k − k2 ≤ examples (with n =1, m = p =0) f (x)=1/x, dom f = R : p⋆ =0, no optimal point • 0 0 ++ f (x)= log x, dom f = R : p⋆ = • 0 − 0 ++ −∞ f (x)= x log x, dom f = R : p⋆ = 1/e, x =1/e is optimal • 0 0 ++ − f (x)= x3 3x, p⋆ = , local optimum at x =1 • 0 − −∞ Convex optimization problems 4–3 Implicit constraints the standard form optimization problem has an implicit -
Metaheuristics1
METAHEURISTICS1 Kenneth Sörensen University of Antwerp, Belgium Fred Glover University of Colorado and OptTek Systems, Inc., USA 1 Definition A metaheuristic is a high-level problem-independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms (Sörensen and Glover, To appear). Notable examples of metaheuristics include genetic/evolutionary algorithms, tabu search, simulated annealing, and ant colony optimization, although many more exist. A problem-specific implementation of a heuristic optimization algorithm according to the guidelines expressed in a metaheuristic framework is also referred to as a metaheuristic. The term was coined by Glover (1986) and combines the Greek prefix meta- (metá, beyond in the sense of high-level) with heuristic (from the Greek heuriskein or euriskein, to search). Metaheuristic algorithms, i.e., optimization methods designed according to the strategies laid out in a metaheuristic framework, are — as the name suggests — always heuristic in nature. This fact distinguishes them from exact methods, that do come with a proof that the optimal solution will be found in a finite (although often prohibitively large) amount of time. Metaheuristics are therefore developed specifically to find a solution that is “good enough” in a computing time that is “small enough”. As a result, they are not subject to combinatorial explosion – the phenomenon where the computing time required to find the optimal solution of NP- hard problems increases as an exponential function of the problem size. Metaheuristics have been demonstrated by the scientific community to be a viable, and often superior, alternative to more traditional (exact) methods of mixed- integer optimization such as branch and bound and dynamic programming. -
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization Professor Dorit S. Hochbaum Contents 1 Introduction 1 2 Formulation of some ILP 2 2.1 0-1 knapsack problem . 2 2.2 Assignment problem . 2 3 Non-linear Objective functions 4 3.1 Production problem with set-up costs . 4 3.2 Piecewise linear cost function . 5 3.3 Piecewise linear convex cost function . 6 3.4 Disjunctive constraints . 7 4 Some famous combinatorial problems 7 4.1 Max clique problem . 7 4.2 SAT (satisfiability) . 7 4.3 Vertex cover problem . 7 5 General optimization 8 6 Neighborhood 8 6.1 Exact neighborhood . 8 7 Complexity of algorithms 9 7.1 Finding the maximum element . 9 7.2 0-1 knapsack . 9 7.3 Linear systems . 10 7.4 Linear Programming . 11 8 Some interesting IP formulations 12 8.1 The fixed cost plant location problem . 12 8.2 Minimum/maximum spanning tree (MST) . 12 9 The Minimum Spanning Tree (MST) Problem 13 i IEOR269 notes, Prof. Hochbaum, 2010 ii 10 General Matching Problem 14 10.1 Maximum Matching Problem in Bipartite Graphs . 14 10.2 Maximum Matching Problem in Non-Bipartite Graphs . 15 10.3 Constraint Matrix Analysis for Matching Problems . 16 11 Traveling Salesperson Problem (TSP) 17 11.1 IP Formulation for TSP . 17 12 Discussion of LP-Formulation for MST 18 13 Branch-and-Bound 20 13.1 The Branch-and-Bound technique . 20 13.2 Other Branch-and-Bound techniques . 22 14 Basic graph definitions 23 15 Complexity analysis 24 15.1 Measuring quality of an algorithm . -
Introduction to Integer Programming – Integer Programming Models
15.053/8 March 14, 2013 Introduction to Integer Programming – Integer programming models 1 Quotes of the Day “Somebody who thinks logically is a nice contrast to the real world.” -- The Law of Thumb “Take some more tea,” the March Hare said to Alice, very earnestly. “I’ve had nothing yet,” Alice replied in an offended tone, “so I can’t take more.” “You mean you can’t take less,” said the Hatter. “It’s very easy to take more than nothing.” -- Lewis Carroll in Alice in Wonderland 2 Combinatorial optimization problems INPUT: A description of the data for an instance of the problem FEASIBLE SOLUTIONS: there is a way of determining from the input whether a given solution x’ (assignment of values to decision variables) is feasible. Typically in combinatorial optimization problems there is a finite number of possible solutions. OBJECTIVE FUNCTION: For each feasible solution x’ there is an associated objective f(x’). Minimization problem. Find a feasible solution x* that minimizes f( ) among all feasible solutions. 3 Example 1: Traveling Salesman Problem INPUT: a set N of n points in the plane FEASIBLE SOLUTION: a tour that passes through each point exactly once. OBJECTIVE: minimize the length of the tour. 4 Example 2: Balanced Partition INPUT: A set of positive integers a1, …, an FEASIBLE SOLUTION: a partition of {1, 2, … n} into two disjoint sets S and T. – S ∩ T = ∅, S∪T = {1, … , n} OBJECTIVE : minimize | ∑i∈S ai - ∑i∈T ai | Example: 7, 10, 13, 17, 20, 22 These numbers sum to 89 The best split is {10, 13, 22} and {7, 17, 20}. -
Integer Optimization Methods for Machine Learning by Allison an Chang Sc.B
Integer Optimization Methods for Machine Learning by Allison An Chang Sc.B. Applied Mathematics, Brown University (2007) Submitted to the Sloan School of Management in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2012 c Massachusetts Institute of Technology 2012. All rights reserved. Author............................................. ................. Sloan School of Management May 18, 2012 Certified by......................................... ................. Dimitris Bertsimas Boeing Leaders for Global Operations Professor Co-Director, Operations Research Center Thesis Supervisor Certified by......................................... ................. Cynthia Rudin Assistant Professor of Statistics Thesis Supervisor Accepted by......................................... ................ Patrick Jaillet Dugald C. Jackson Professor Department of Electrical Engineering and Computer Science Co-Director, Operations Research Center 2 Integer Optimization Methods for Machine Learning by Allison An Chang Submitted to the Sloan School of Management on May 18, 2012, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Operations Research Abstract In this thesis, we propose new mixed integer optimization (MIO) methods to ad- dress problems in machine learning. The first part develops methods for supervised bipartite ranking, which arises in prioritization tasks in diverse domains such as infor- mation retrieval, recommender systems, natural language processing, bioinformatics, and preventative maintenance. The primary advantage of using MIO for ranking is that it allows for direct optimization of ranking quality measures, as opposed to current state-of-the-art algorithms that use heuristic loss functions. We demonstrate using a number of datasets that our approach can outperform other ranking methods. The second part of the thesis focuses on reverse-engineering ranking models. -
Constrained Optimization 5
Constrained Optimization 5 Most problems in structural optimization must be formulated as constrained min- imization problems. In a typical structural design problem the objective function is a fairly simple function of the design variables (e.g., weight), but the design has to satisfy a host of stress, displacement, buckling, and frequency constraints. These constraints are usually complex functions of the design variables available only from an analysis of a finite element model of the structure. This chapter offers a review of methods that are commonly used to solve such constrained problems. The methods described in this chapter are for use when the computational cost of evaluating the objective function and constraints is small or moderate. In these meth- ods the objective function or constraints these are calculated exactly (e.g., by a finite element program) whenever they are required by the optimization algorithm. This approach can require hundreds of evaluations of objective function and constraints, and is not practical for problems where a single evaluation is computationally ex- pensive. For these more expensive problems we go through an intermediate stage of constructing approximations for the objective function and constraints, or at least for the more expensive functions. The optimization is then performed on the approx- imate problem. This approximation process is described in the next chapter. The basic problem that we consider in this chapter is the minimization of a function subject to equality and inequality constraints minimize f(x) such that hi(x) = 0, i = 1, . , ne , (5.1) gj(x) ≥ 0, j = 1, . , ng . The constraints divide the design space into two domains, the feasible domain where the constraints are satisfied, and the infeasible domain where at least one of the constraints is violated. -
Tabu Search for Constraint Satisfaction
Tabu Search for Constraint Solving and Its Applications Jin-Kao Hao LERIA University of Angers 2 Boulevard Lavoisier 49045 Angers Cedex 01 - France 1. Introduction The Constraint Satisfaction Problem (CSP) is a very general problem able to conveniently formulate a wide range of important applications in combinatorial optimization. These include well-known classical problems such as graph k-coloring and satisfiability, as well as many practical applications related to resource assignments, planning and timetabling. The feasibility problem of Integer Programming can also be easily represented by the CSP formalism. As elaborated below, we provide case studies for specific CSP applications, including the important realm of frequency assignment, and establish that tabu search is the best performing approach for these applications. The Constraint Satisfaction Problem is defined by a triplet (X, D, C) where: – X = {x1,· · ·,xn} is a finite set of n variables. – D = {Dx1 ,· · ·,Dxn} is a set of associated domains. Each domain Dxi specifies the finite set of possible values of the variable xi. – C = {C1,· · ·,Cp} is a finite set of p constraints. Each constraint is defined on a set of variables and specifies which combinations of values are compatible for these variables. Given such a triplet, the basic problem consists in finding a complete assignment of the values to the variables such that that all the constraints are satisfied. Such an assignment is called a solution to the given CSP instance. Since the set of all assignments is given by the Cartesian product Dx1 ×· · ·×Dxn, solving a CSP means to determine a particular assignment among a potentially huge search space. -
Integer and Combinatorial Optimization
Integer and Combinatorial Optimization Karla L. Hoffman Systems Engineering and Operations Research Department, School of Information Technology and Engineering, George Mason University, Fairfax, VA 22030 Ted K. Ralphs Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015 Technical Report 12T-020 Integer and Combinatorial Optimization Karla L. Hoffman∗1 and Ted K. Ralphsy2 1Systems Engineering and Operations Research Department, School of Information Technology and Engineering, George Mason University, Fairfax, VA 22030 2Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015 January 18, 2012 1 Introduction Integer optimization problems are concerned with the efficient allocation of limited resources to meet a desired objective when some of the resources in question can only be divided into discrete parts. In such cases, the divisibility constraints on these resources, which may be people, machines, or other discrete inputs, may restrict the possible alternatives to a finite set. Nevertheless, there are usually too many alternatives to make complete enumeration a viable option for instances of realistic size. For example, an airline may need to determine crew schedules that minimize the total operating cost; an automotive manufacturer may want to determine the optimal mix of models to produce in order to maximize profit; or a flexible manufacturing facility may want to schedule production for a plant without knowing precisely what parts will be needed in future periods. In today's changing and competitive industrial environment, the difference between ad hoc planning methods and those that use sophisticated mathematical models to determine an optimal course of action can determine whether or not a company survives. -
Linear Programming: Geometry, Algebra and the Simplex Method
ISyE 3133 Lecture Notes ⃝c Shabbir Ahmed Linear Programming: Geometry, Algebra and the Simplex Method A linear programming problem (LP) is an optimization problem where all variables are continuous, the objective is a linear (with respect to the decision variables) function , and the feasible region is defined by a finite number of linear inequalities or equations. LP1 is possibly the best known and most frequently used branch of optimization. Beginning with the seminal work of Dantzig2, LP has witnessed great success in real world applications from such diverse areas as sociology, finance, transportation, manufacturing and medicine. The importance and extensive use of LPs also come from the fact that the solution of certain optimization problems that are not LPs, e.g., integer, stochastic, and nonlinear programming problems, is often carried out by solving a sequence of related linear programs. In this note, we discuss the geometry and algebra of LPs and present the Simplex method. 1.1 Geometry of LP Recall that an LP involves optimizing a linear objective subject to linear constraints, and so can be written in the form f > > ≤ g min c x : ai x bi i = 1; : : : ; m : An LP involving equality constraints can be written in the above form by replacing each equality constraint by two inequality constraints. The feasible region of an LP is of the form f 2 Rn > ≤ g X = x : ai x bi i = 1; : : : ; m which is a polyhedron (recall the definitions/properties of hyperplanes, halfspaces, and polyhedral sets). Thus an LP involves minimizing a linear function over a polyhedral set. -
Data-Driven Satisficing Measure and Ranking
Data-driven satisficing measure and ranking Wenjie Huang ∗ July 3, 2018 Abstract We propose an computational framework for real-time risk assessment and prioritizing for random outcomes without prior information on probability distributions. The basic model is built based on sat- isficing measure (SM) which yields a single index for risk comparison. Since SM is a dual representation for a family of risk measures, we consider problems constrained by general convex risk measures and specifically by Conditional value-at-risk. Starting from offline optimization, we apply sample average approximation technique and argue the convergence rate and validation of optimal solutions. In online stochastic optimization case, we develop primal-dual stochastic approximation algorithms respectively for general risk constrained problems, and derive their regret bounds. For both offline and online cases, we illustrate the relationship between risk ranking accuracy with sample size (or iterations). Keywords: Risk measure; Satisficing measure; Online stochastic optimization; Stochastic approxima- tion; Sample average approximation; Ranking; 1 Introduction Risk assessment is the process where we identify hazards, analyze or evaluate the risk associated with that hazard, and determine appropriate ways to eliminate or control the hazard. Risk assessment techniques have been widely applied in many area including quantitative financial engineering (Krokhmal et al. 2002), health and environment study (Zhang and Wang 2012; Van Asselt et al. 2013), transportation science (Liu et al. 2017), etc. Paltrinieri et al. (2014) point out that traditional risk assessment methods are often limited by static, one-time processes performed during the design phase of industrial processes. As such they often use older data or generic data on potential hazards and failure rates of equipment and processes and cannot be easily updated in order to take into account new information, giving a more complete view of the related risks. -
Constrained Quadratic Optimization: Theory and Application for Wireless Communication Systems
Constrained Quadratic Optimization: Theory and Application for Wireless Communication Systems by Erik Hons A thesis presented to the University of Waterloo in ful¯lment of the thesis requirement for the degree of Master of Applied Science in Electrical and Computer Engineering Waterloo, Ontario, Canada, 2001 c Erik Hons 2001 ° I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. I further authorize the University of Waterloo to reproduce this thesis by pho- tocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. ii The University of Waterloo requires the signatures of all persons using or pho- tocopying this thesis. Please sign below, and give address and date. iii Abstract The suitability of general constrained quadratic optimization as a modeling and solution tool for wireless communication is discussed. It is found to be appropriate due to the many quadratic entities present in wireless communication models. It is posited that decisions which a®ect the minimum mean squared error at the receiver may achieve improved performance if those decisions are energy constrained. That theory is tested by applying a speci¯c type of constrained quadratic optimization problem to the forward link of a cellular DS-CDMA system. Through simulation it is shown that when channel coding is used, energy constrained methods typi- cally outperform unconstrained methods. Furthermore, a new energy-constrained method is presented which signi¯cantly outperforms a similar published method in several di®erent environments. -
Linear & Quadratic Knapsack Optimisation Problem
linear & quadratic knapsack optimisation problem Supervisor: Prof. Montaz Ali Alex Alochukwu, Ben Levin, Krupa Prag, Micheal Olusanya, Shane Josias, Sakirudeen Abdulsalaam, Vincent Langat January 13, 2018 GROUP 4: Graduate Modelling Camp - MISG 2018 Introduction ∙ The Knapsack Problem is considered to be a combinatorial optimization problem. ∙ The best selection and configuration of a collection of objects adhering to some objective function defines combinatorial optimization problems. 1 Problem Description: Knapsack Problem To determine the number of items to include in a collection such that the total weight is less than or equal to a given limit and the total value is maximised. 2 Problem Description: Knapsack Problem 3 Mathematical Formulation:Linear Knapsack problem Given n-tuples of positive numbers (v1; v2; :::; vn) , (w1; w2; :::; wn) and W > 0. The aim is to determine the subset S of items each with values,vi and wi that Xn Maximize vixi; xi 2 f0; 1g; xi is the decision variable (1) i=1 Xn Subject to: wixi < W; (2) i=1 Xn where W < wi: i=1 4 Quadratic Knapsack Problem ∙ Extension of the linear Knapsack problem. ∙ Additional term in the objective function that describes extra profit gained from choosing a particular combination of items. 5 Mathematical Formulation Xn Xn−1 Xn Maximize cixi + dijxixj (3) i=1 i=1 j=i+1 Xn Subject to: wjxj < W; j = f1; 2; :::; ng; (4) j=1 Xn where x 2 f0; 1g; Max wj ≤ W < wj: j=1 6 Application A Knapsack model serves as an abstract model with broad spectrum applications such as: ∙ Resource allocation problems ∙ Portfolio optimization ∙ Cargo-loading problems ∙ Cutting stock problems 7 Complexity Theory ∙ The abstract measurement of the rate of growth in the required resources as the input n increases, is how we distinguish among the complexity classes.