Lecture 1 Stochastic Optimization: Introduction

1 Lecture 1 Stochastic Optimization: IntroductionJanuary 8, 2018 Uday V. ShanbhagLecture 1 Optimization Concerned with mininmization/maximization of mathematical functions Often subject to constraints Euler (1707-1783):Nothing at all takes place in the universe in whichsome rule of the maximum or minimum does not apply. Important tool in the analysis/design/control/simulation of physical,economic, chemical and biological systems Model apply algorithm check solutionStochastic Optimization1 Uday V. ShanbhagLecture 1 Unconstrained optimizationUnconstrainedminimizex Rnf(x) Xis defined asX,Rn Examples:f(x) =x3 3x2.

2 Important application: Data fitting and regressionStochastic Optimization2 Uday V. ShanbhagLecture 1 Unconstrained optimization: An exampleGiven a data set{yi,xi1,..,xip}np=1(nrecords, with the dependentvariableyiand independent variablexi1,..,xip).The linear regression model assumes that the relationship between thedependent variableyand the independent variablesxiis is captured as follows:yi=xi0+p j=1 pxip+ i, i= 1,..,nwhere idenotes a random variable. More compactly, we may state this asfollows:y=X + ,wherey, ,X, xT1 ..xTn . Stochastic Optimization3 Uday V. ShanbhagLecture 1 Then the least-squares estimator is defined as follows: =argmin X y Optimization4 Uday V.

3 ShanbhagLecture 1 Convex optimizationConvexminimizex Rnf(x)subject tox X,whereXis a convex set andfis a convex 1 (Convexity of sets and functions) A setX Rnis a convex set ifx1,x2 Xthen( x1+ (1 )x2) Xfor all [0,1]. A functionfis said to be convex iff( x1+ (1 )x2) f(x1) + (1 )f(x2), [0,1]. Stochastic Optimization5 Uday V. ShanbhagLecture 1 A functionfis said to be strictly convex iff( x1+ (1 )x2)< f(x1) + (1 )f(x2), [0,1]. A functionfis said to be strongly convex with parameter iff( x1+(1 )x2) f(x1)+(1 )f(x2) 12 (1 ) x1 x2 2, [0,1].Note that in the above definitionfdoes not need to be 2 (Convexity of differentiable functions)Consider a differ-entiable functionf:Rn R.

4 A functionfis said to be convex iff(x2) f(x1) + xf(x1)T(x2 x1), x1,x2 Optimization6 Uday V. ShanbhagLecture 1 A functionfis said to be strongly convex with parameter if( xf(x1) xf(x2))T(x1 x2) x1 x2 2, x1,x2 Rn. Any local solution of (Convex) is a global solution Examples of convex constraints:X,{x:Ax=b,x 0} quadratic constraints:X,{x: Ni=1(xi ai)2 b}. Examples of convex (x) = (x) =12xTQx+cTx,whereQ 0. Application: Controller design, constrained least-squares, Optimization7 Uday V. ShanbhagLecture 1 Nonlinear programNLPminimizex Xf(x) f:Objective function is a possibly nonconvex function x Rn: Decision variables X Rnis a possibly nonconvex set f:X R Applications: Nonlinear regression, process control in chemical engineer-ing, etc.

5 : Stochastic Optimization8 Uday V. ShanbhagLecture 1 Discrete optimizationDiscreteminimizex Rnf(x)subject tox Z. Zis a finite set implying thatxcan take on discrete values {0,1}. Sometimesx1 R,x2 {0,1}; the resulting problem is called amixed-integer problem Applications: facility location problems, unit commitment problemsStochastic Optimization9 Uday V. ShanbhagLecture 1 Convex optimization relevance in this course Stochastic optimization captures a broad class of problems, includingconvex, nonconvex (time permitting), and discrete optimization problems(not considered here). In this course, we focus on the following: Convexstochastic optimization problems (including Stochastic pro-grams with recourse) Monotonestochastic variational inequality problems (subsumes stochas-tic convex optimization and captures Stochastic Nash games, stochas-tic contact problems, Stochastic traffic equilibrium problems) Robustoptimization problems Applications:Statistical learning problems Convexity is crucial and will be leveraged extensively during the course!

6 ! Stochastic Optimization10 Uday V. ShanbhagLecture 1 Problems complicated by uncertainty In the aforementioned (deterministic) problems, parameters are knownwith certainty. Specifically, given a functionf(x; ), we consider twopossibilities: is a random variable. Our focus is then on solving the following:minx XE[f(x, )](Stoch-Opt) is unavailable and instead we have that U(whereUis anuncertainty set). A problem of interest is then:minx Xmax Uf(x, )(Robust-Opt) Stochastic Optimization11 Uday V. ShanbhagLecture 1 We motivate this line of questioning by considering the classical newsven-dor problemStochastic Optimization12 Uday V.

7 ShanbhagLecture 1A short detour Probability Spaces Throughout this course, we will be utilizing the notion of a probabilityspace( ,F,P). This mathematical construct captures processes (either real or synthetic)that are characterized by randomness. This space is constructed for a particular such process and on everyoccasion this process is examined, both the set of outcomes and theassociated probabilities are the same. The sample-space is a nonempty set that denotes the set ofoutcomes. This represents a single execution of the experiment. The -algebraFdenotes the set of events where each event is a setcontaining zero or more Optimization13 Uday V.

8 ShanbhagLecture 1 The assigmnent of probabilities to the events is captured byP. Once the space( ,F,P)is established, thennatureselects an outcome from . As a consequence, all events that contain as one of itsoutcomes are said to have occurred. If nature selects outcomes infinitely often, then the relative frequenciesof occurrence of a particular event corresponds with the value specifiedby the probability Optimization14 Uday V. ShanbhagLecture 1 Properties ofF: F Fclosed under complementation:A F= ( \A) F. Fis closed under countable unionsAi Ffori= 1, ,impliesthat( i=1Ai) F.

9 Properties ofP. The probability measureP:F [0,1]such thatPis Pis countably additive: If{Ai} i=1 Fdenotes a countable col-lection of pairwise disjoint sets (Ai Aj= fori6=j), thenP( i=1Ai) = i=1P(Ai). The measure of the sample-space is one orP( ) = Optimization15 Uday V. ShanbhagLecture 1A short detour Probability Spaces: II Example coin toss ,{H,T}. The algebraFcontains22= 4eventsF,{{},{H},{T},{H,T}}. Furthermore,P({}) = 0,P({H}) = ,P({T}) = , andP({H,T}) = 1. Example coin toss ,{HH,HT,TH,TT}. Stochastic Optimization16 Uday V. ShanbhagLecture 1 The algebraFcontains24= 16eventsF,{{},{HH},{TT},{HT},{TH},{HH,TT },{HH,HT},{HH,TH}{HT,TT},{HT,TH},{TH,TT} ,{HH,HT,TH},{HH,HT,TT},{HH,TH,TT},{HT,TH ,TT}{HH,TH,HT,HH}}.

10 Furthermore,P({}) = 0,P(A1) = ,P(A2) = ,P(A3) = Stochastic Optimization17 Uday V. ShanbhagLecture , andP({HH,TH,HT,HH}) = 1, whereA1 {HH,HT,TH,TT}A2 {{HH,TT},{HH,HT},{HH,TH},{HT,TT},{HT,TH} ,{TH,TT},A3 {{HH,HT,TH},{HH,HT,TT},{HH,TH,TT},{HT,TH ,TT}}, Stochastic Optimization18 Uday V. ShanbhagLecture 1 Random variables Given a probability space( ,F,P), a random variable represents afunction on a sample-space with measurable values. Specifically,Xis a random variable defined asX: E, whereEis a measurable space. Consequently,P(X S) =P( |X( ) S). ( )as follows:X( ) = 100, =H 100, = Optimization19 Uday V.}