Optimization Theory and Applications

Xiaoxi Li

EMS & IAS, Wuhan University

Spring, 2017

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 1 / 15

Organization of this lecture

Contents:Introduction to the course Optimization Theory and Applications.Some motivating examples of optimization.

Application 1: Molecular Biology.Application 2: Finance.Application 3: Meteorology.Application 4: Artificial Intelligence.

Some mathematical preliminaries.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 2 / 15

Course information

Basic information

Title.Optimization Theory and ApplicationVolume. 3 course hours by 18 weeksTime and venue. Friday 9:50-12:15 at room 01-4-202.

Grading rule:final grade ≈ 20% HW + 25% mid-term exam + 55% final examCourse description. This is an introduction course in optimizationtheory and its applications, containing the following two parts:

Nonlinear Optimization: unconstrained and constrainedoptimization.Dynamic Optimization: dynamic programming, calculus ofvariation, optimal control.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 3 / 15

Diversified models of optimization

Compare with other related (optimization) courses at IAS:Operations Research (S5): linear programming, networkoptimization, integer programming (models featured a discretestructure), markovian decision processes, queuing theory(stochastic models).Control Theory (S5): focus on dynamic optimization incontinuous time: calculus of variations and optimal control ;Dynamic Programming (S6): dynamic optimization in discretetime with oriented for macroeconomic applications.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 4 / 15

Diversified models of optimization (cont.)

A generic optimization problem: minx∈X f (x), where x is the decisionvariable, f (x) is the cost function, and X ⊆ Rn is the choice set.

Categories of optimization models:constrained (v.s. unconstrained) optimization: X 6= Rn.

discrete (v.s. continuous) optimization: X is a finite set.

nonlinear (v.s. linear) optimization: "x ∈ X" can be written as g(x)≤ 0,h(x) = 0and either f (·) or g(·) is nonlinear.

stochastic (v.s. deterministic) optimization: f (x) = Ew[F(x,w)] with w random.

dynamic (v.s. static) programming: can be modeled as a complex stochasticoptimization problem (multi-stage) where

information about w is revealed stage by stage;decisions are also made by stages and make use of the availableinformation;some "different" methodology is employed.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 5 / 15

Some application examples: nonlinear optimization

Molecular Biology. Nature optimizes.

An important problem in biochemistry, for example in pharmacology, isto determine the geometry of a molecule. Various techniques arepossible (X-ray crystallography, nuclear magnetic resonance,. . . ) oneof these is convenient when

the chemical formula of the molecule is knownthe molecule is not available, making it impossible to conduct anyexperiment,one has some knowledge of its shape and one wants to refine it.

The idea is then to compute the positions of the atoms in the spacethat minimize the associated potential energy.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 6 / 15

Application 1: Molecular Biology (cont.)

Let N be the number of atoms and call xi ∈ R3 the spatial position ofthe i-th atom. To the vector X = (x1, ...,xN) ∈ R3N is associated apotential energy f (X) (the conformational energy"). The problem is:

minX=(x1,...,xN)∈R3N

f (X) = ∑i,j

Lij(xi,xj)+∑i,j

Vij(xi,xj)+∑i,j,k

Aijk(xi,xj,xk)+others

where L, V, A are sorts of energies among atoms:Lij(xi,xj) = λij

(‖xi− xj‖−dij

)is the Bond length energy;

Vij(xi,xj) = vij

(δij

‖xi−xj‖

)6−wij

(δij

‖xi−xj‖

)12is a Van der Waals energy;

Aijk(xi,xj,xk) = αijk(θijk− θijk

)2 is the Valence angle energy. (θijk isthe angle formed by atoms i, j,k’s positions and dij, λij, vij, δij,wij,αij, θijk are known atoms-involved parameters).

Other types of energies may also be considered: electrostatic, torsionangles, etc.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 7 / 15

Application 2: Finance (Markowitz Theory ofMean-Variance Optimization)

Finance (Mean-Variance Optimization). Human optimize.

There are N financial assets.µ = (µ1, ...,µN) is the expected return vector, and σ = (σ1, ...,σN) isthe vector of standard deviation of the assets.Σ = (σij) is the n×n symmetric covariance matrix.p = (p1, ...,pN) is the price vector and w is the available wealth.Let x = (x1, ...,xN) be the vector of purchased asset amounts (theportfolio).

The problem is to minimize the portfolio variance while yielding a targetvalue R of expect return:

minx

xTΣx

s.t. µTx≥ R

pTx = w

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 8 / 15

Application 2: Finance (cont.)

There are many other applications of optimization methods in finance,such as:

Linear Optimization: Asset/Liability Cash Flow Matching, AssetPricing and ArbitrageNonlinear Optimization: Volatility EstimationInteger Optimization: Constructing an Index FundDynamic Optimization: Option Pricing, Structuring Asset BackedSecuritiesStochastic Optimization: Value-at-Risk, Corporate DebtManagement

For more detail, please refer to

G. Cornuejols and R. Tutuncu. Optimization Methods in Finance.Cambridge University Press, 2006.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 9 / 15

Application 3: Meteorology

Meteorology. Optimization as an indirect tool.

To forecast the weather is to know the state of the atmosphere inthe future. This is quite possible, at least theoretically (and withinlimits due to the chaotic character of phenomena involved).Let p(z, t) be the state of the atmosphere at point z ∈ R3 and timet ∈ [0,7] (assuming a forecast over one week, say); p is actually avector made up of pressure, wind speed, humidity . . .The evolution of p along time can be modeled (via fluid mechanics)

∂p∂ t

(z, t) = Φ(p(z, t)

),

where Φ is a certain differential operator.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 10 / 15

Application 3: Meteorology (cont.)

The available information contains all the meteorologicalobservations collected in the past, say during the preceding day.Let us denote by Ω = ωii∈I these observations, where each ωi

represents the value of p at a certain point (zi, ti).An optimization can be roughly defined defined as:

minp(·,·)

∑i∈I

d(p(zi, ti),ωi

)s.t

∂p∂ t

(z, t) = Φ(p(z, t)

), (z, t) ∈ R3× [−1,7].

Here the decision variable p(·, ·) is a function defined onR3× [−1,7]. This is an dynamic optimization problem in continuousand also an optimization problem in infinite dimensional space;d is a distance function that needs to be defined appropriately.The dynamic problem can be (by approximation) transformed to aNLP by discretization in the continuous state. This is also usuallyhow this type of problems have been numerically solved.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 11 / 15

Application 4: Artificial Intelligence

Artificial Intelligence. Robots optimize.

For games of finite steps with perfect information (all chessboard gamesfor example), it’s possible to use the backward induction (dynamicprogramming) algorithm to calculate the value v∗(s) (to win or not) andthe optimal policy p∗(s) at each state s (chessboard configuration).

However, such a search tree contains approximately bd possiblesequence of moves, where ("the curse of dimensionality")

b is the game’s breadth (number of legal moves per position),d is its depth (game length).

In large games, such as chess (b≈ 35, d ≈ 80) and especially Go(b≈ 250, d ≈ 150), exhaustive search is infeasible

Google’s AlghaGo uses a combination algorithm of reinforcementlearning (= approximate dynamic programming) and of supervisedlearning to human being.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 12 / 15

Application 4: Artificial Intelligence (cont.)

Search tree of the large game GO.

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 13 / 15

Application 4: Artificial Intelligence (cont.)

How does AlghaGo work? Here is a rough explanation.

Use historical data to mimic human expert’s policy (supervised learningalgorithm), turn a zero-sum game into a dynamic programming problem;

To solve the curses of dimensionality: at each s,

Bellman equation: vn+1(s) = maxa[f (s,a)+∑s′ `(s′|s,a)vn(s′)

].

breadth concern: instead of computing all possibilities, samplerandomly to choose one move according to p(·|s).depth concern: for each path of long steps, instead of going to theend to obtain the winning probability, only limited steps are carriedout, in addition to an auxiliary value function v(s).improvement : AlphaGo simulates frequently the play against itselfso as to revise v(s), which in turn helps revise p(·|s). Designedproperly, v(s) will converge to v∗(s) and p(·|s) to p(s).

Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 14 / 15

Application 4: Artificial Intelligence (cont.)

References:

David Silver, etc. Mastering the game of Go with deep neural networksand tree search. Nature 529, 484-489, 2016.

Warren B. Powell. Approximate Dynamic Programming: solving thecurses of dimensionality, 2e. (c) John Wiley and Sons, 2011.Introduction (Li, X.) Optimization Theory and Applications Spring, 2017 15 / 15