robust optimization.pdf

8/14/2019 Robust Optimization.pdf

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Applications of Robust Linear &Nonlinear Optimization in Engineering

Design

Masters Thesis

Mechanical Engineering

by

Harshal Avinash Mungikar

Graduate Advisor: Dr. Jagannatha Rao1


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Abstract

Optimization problems often characterized by uncertain datain real world

Robust optimization immunizes solution from data uncertain-ties

Demonstrated applicability of robust optimization methods byBen-Tal & Nemirovski (Linear) & Zhang (Nonlinear) to engg.case studies

Suggested modications to improve methodologies

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Objectives

Objective 1: Demonstrate applicability of Robust linear (Ben-Tal & Nemirovski) & Robust nonlinear (Zhang Y.) method-ologies to general engineering case studies & advantages overtraditional methods

Objective 2: Analyze advantages & limitations of method-ologies using case studies

Objective 3: Suggest certain modications in methodologiesto either improve the scope of applicability or to provide betteroptimal objective value

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Methodology of Optimization

Optimization method to nd best solution from a feasibleset

minimize f 0 ( x )

subject to f i ( x ) bi , i = 1 , ...., m(1)

f 0 : R n

R is objective function, f i : R n

R are constraints

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Reasons for Uncertainty

Measurement process variations

Outer environment variations (outside the system)

Inner environment variations (inside the system)

Variations in different variations of same product

Uncertainty in design process

Implementation errors5


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Major Advances in Robust Optimization

Stochastic Optimization: Soft - Constrained Problems

Require uncertain data to be truly random not always

Probability distribution exactly known cannot be sure al-ways

Guarantee of constraint satisfaction acceptable violationwith penalty

Counterpart computationally tractable

difficult to verify

Sensitivity Analysis:Post-analysis tool giving stability of optimal solution under pertur-bations

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Soysters work (1973): feasible region constraint restrict-ing sum of activity sets in a resource set (Ultraconservative)

Falks work (1976): uncertainty in objective vector c C gave conditions for c (optimal of c) to give solution usingSoysters analysis

Thuente (1980): Generelized linear programming

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Singh (1982): Extension of GLP to convex programs

Mulvey et. al. (1995): Robust Optimization, problem datausing set of scenarios (tired combining solution robustness& model robustness). Trade-off between the two usedpenalty functions to minimize violations (which are denitelyexpected)

Ben-Tal & Nemirovski (1997 - 2002): Robust convex op-timization, robust solutions to linear programming

Gurav et.al. (2004): Uncertianty-based design optimization

Kogiso et. al. (2008): Robust Topology Optimization8


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Robust Linear Optimization

General form: Linear optimization problem( P ) min {cT x | Ax 0 , f T x = 1 }

Uncertain LP: family of LP instancesmin {c

T x | Ax 0 , f

T x = 1 }AU

Robust counterpart:( P U ) min {cT x | Ax 0 , A U, f T x = 1 }

U assumed to be closed & convex.

min {cT x | [r(0)i ]

T x ||R T i x||, i = 1 , . . . , m ; f T x = 1 } (2)9


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Ellipsoidal Uncertainty Set

U = {( u ) | ||Qu || 1} (3)u ( u ) is the affine embedding from R L to R K , Q being an M Lmatrix

L = M < K, Q non singular at ellipsoid (partial uncer-tainty)

Sum of at ellipsoid & linear subspace Ellipsoidal cylinder

L = M = K, Q non singular standard K-dimensional ellip-soid

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RC Derivation Explanation

U = {A = P 0 +k

j =1

u j P j |u T u 1}where P j , j = 0 , . . . , k are m n matrices

r( j )i ith row of P j matrix

R i n K matrix (columns = rows r( j )i )

ith row of ( u ) r(0)i + R i u

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RC Derivation Explanation contd.

x R n is robust feasible f T x = 1

[r(0)i ]

T

x + ( R i u )T

x 0 , ( u, ||u || 1) [i = 1 , . . . , m ]

(R i u ) T x = u T R T i x = ||R T i x|| when ||u || 1

Robust Counterpart : (Conic Quadratic)

min {cT x | [r(0)i ]

T x ||R T i x||, i = 1 , . . . , m ; f T x = 1 } (4)

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Interval Model of Uncertainty

Every uncertain data runs through independent intervals (allworst values considered at a time)

Uncertain problem instances

min x cT x : Ax b|c j cn j | c j

|A ij A nij | A ij|bi bni | b i

(5)

Robust Counterpart:min

x,y j [cn j x j + c j y j ] : j [A

nij x j + A ij y j ] bi b i ,

y j x j y j , (6)

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Ellipsoidal Model of Uncertainty

Cases when all uncertain data do not have their worst valuesat a time (eg.: stochastic data)

Consider linear inequality:a 0 + n j =1 a j x j 0 (7)

coefficients of constraint a = ( a 0 , a 1 ,...,a n ) T are random

Robust inequality (ellipsoidal uncertainty)a n0 +

n j =1 a

n j x j + (1 , x T ) V (1 , x T ) T 0 ,

V = E {( a a n )( a a n ) T }(8)

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Case Study: Inventory UtilizationProblem

Cabinet type (x 1 ... 4 ) Wood (W 1 ... 4 ) Labor (L 1 ... 4 ) Revenue (R 1 ... 4 )Bookshelf 10 2 100

With Doors 12 4 150With Drawers 25 8 200

Custom 20 12 400

Objective: Maximize weekly revenue by optimizing each prod-uct qty.

Constraints: Max. & Min. qty.s of wood and labor inven-tory.

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Problem Formulation & Nominal Result

Cabinet type (x 1 ... 4 ) Wood (W 1 ... 4 ) Labor (L 1 ... 4 ) Revenue (R 1 ... 4 )Bookshelf 10 2 100

With Doors 12 4 150With Drawers 25 8 200

Custom 20 12 400

maxx i

R i x i :500 W i x i 5000 , i = 1 , ..., 4200 L i x i 1500 , i = 1 , ..., 4

x i 0 , i = 1 , ..., 4(9)

Max. Revenue x1 x2 x3 x462300 375 0 0 62

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Uncertainty & Consequences:

Assuming uncertainty of (2 , 1 , 2 , 2) T in wood

Labor uncertainty of (1 , 1 , 2 , 2) T

Max. constraint violation found 33 %

Can cause problem solution to become practically meaningless

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Interval Model of Uncertainty

maxx i

R i x i :

500 ( W i + wi W i ) x i 5000 , i = 1 , ..., 4 , wi = [ wi , wi ]200 ( L i + li L i ) x i 1500 , i = 1 , ..., 4 , li = [ li , li ]

x i 0 , i = 1 , ..., 4(10)

Max. Revenue x1 x2 x3 x447800 362 0 0 29

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Ellipsoidal Model of Uncertainty

maxx i

R i x i :

500 W i x i w x T V w x, i = 1 , ..., 4 ,W i x i + w

x T V w x 5000 , i = 1 , ..., 4 ,

200 L i x i l x T V lx, i = 1 , ..., 4 ,L i x i + l x T V lx 1500 , i = 1 , ..., 4 ,

x i 0 , i = 1 , ..., 4

(11)

Max. Revenue x1 x2 x3 x453350 193 227 0 0

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Results & Discussion

Max. RevenueNominal

Max. RevenueInterval

Max. RevenueEllipsoidal

62300 47800 53350

Small perturbations in the problem data could make solutionpractically meaningless

Ellipsoidal model provides better optimal value compared tointerval model for same level of uncertainty chosen

As the uncertainty level increases, the difference between el-lipsoidal and interval optimals increases

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Robust Nonlinear Optimization

Previous work by Ben-Tal & Nemirovski dealt with only linearproblems

Uncertain data should be linear functions

Previous work applicable only to inequality constrained prob-lems

Y. Zhang (2007) proposed methodology for linear & nonlin-ear problems with moderate uncertainty & provides solutionsrobust to the rst-order

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Formulation:

min y,uU

s.t.

( y,u, s )

F ( y,u, s ) = 0 ,

( F y ys + F s ) = 0 ,

gi ( y,u, s ) + eT i ( G y ys + G s ) D q 0 , i = 1 , . . . , m (12)

(y, u, s ) state variable, design variable & nominal value of uncertain parameter respectively

F, gi , system state equation, safety constraint & uncer-tainty magnitude (L-1 norm)

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Formulation:

Assumption: safety constraint strictly satisable

Uncertainty set: S := {s + D : p 1}

Taylors approx.: to linearize G in neighborhood of sgi ( u, s + D ) gi ( u, s ) + s gi ( u, s ,D

Holders inequality:

|c, x

| x p c q for 1

p + 1

q = 1 , 1

p, q + max s

S gi ( u, s ) gi ( u, s ) + D T s gi ( u, s ) q 0 (13)

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Robust Solution Characteristics:

First-order robust the rst-order approx. of constraint inthe desired neighborhood of s used to calculate robust optimal

No guarantee that gi < 0

But max. constraint violation capped by term L2 2

L max. rate of change of sensitivity of gi w.r.t. s

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Case Study 1: Building Cable Length Problem

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Problem Formulation:

mins.t.

f : 4i=1 w i ( x i x 0 ) 2 + ( y i y0 ) 2( x 1 1) 2 + ( y1 4) 2 4;( x 2 9) 2 + ( y2 5) 2 1;2 x 3 4; 3 y3 1;6 x 4 8; 2 y4 2

(14)

Building weights: w = [1 .5 , 1 , 1 .25 , 0 .5] T

Objective Function: Minimize total weighted length of wirefrom common point to individual buildings

Constraints: Building receiving point should lie on or insidebuilding circumference

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Nominal Solution:

Weighted Sum of

Dis-tances

( x 0 , y 0 ) ( x 1 , y 1 ) ( x 2 , y 2 ) ( x 3 , y 3 ) ( x 4 , y 4 )

11.9745 (4.17,2.11)

(2.72,2.97)

(8.14,4.48)

(4, -1) (6, 2)

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Robust Formulation:

min

s.t.

f + f w q

( x 1 1) 2 + ( y1 4) 2 4;( x 2 9) 2 + ( y2 5) 2 1;2 x 3 4; 3 y3 1;6 x 4 8; 2 y4 2

(15)

Uncertainty magnitude: = [0 .1 , 0 .08 , 0 .25 , 0 .08] T (if takenas individual uncertainty) and = 1 = 0.51 (according tothe L-1 norm)

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q=1 Case Optimal Solution:

Weighted Sum of

Dis-tances

( x 0 , y 0 ) ( x 1 , y 1 ) ( x 2 , y 2 ) ( x 3 , y 3 ) ( x 4 , y 4 )

17.5670 (4.81,2.07)

(2.78,3.1)

(8.18,4.42)

(4, -1) (6, 2)

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q=2 Case Optimal Solution:

Weighted Sum of

Dis-tances

( x 0 , y 0 ) ( x 1 , y 1 ) ( x 2 , y 2 ) ( x 3 , y 3 ) ( x 4 , y 4 )

15.018 (4.58,2.09)

(2.76,3.06)

(8.16,4.45)

(4, -1) (6, 2)

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q= Case Optimal Solution:

Weighted Sum of

Dis-tances

( x 0 , y 0 ) ( x 1 , y 1 ) ( x 2 , y 2 ) ( x 3 , y 3 ) ( x 4 , y 4 )

14.1243 (4.97,2.39)

(2.85,3.25)

(8.16,4.45)

(4, -1) (6, 2)

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Modied Robust Formulation:

min

s.t.

f + , f w

( x 1 1) 2 + ( y1 4) 2 4;( x 2 9) 2 + ( y2 5) 2 1;

2 x 3 4; 3 y3 1;6 x 4 8; 2 y4 2

(16)

Uncertainty: = [0 .1 , 0 .08 , 0 .25 , 0 .08] T

% uncertainty = [6 .7% , 8% , 20% , 16%] T

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Modied Case Optimal Solution:

Weighted Sum of

Dis-tances

( x 0 , y 0 ) ( x 1 , y 1 ) ( x 2 , y 2 ) ( x 3 , y 3 ) ( x 4 , y 4 )

13.4164 (4.27,1.81)

(2.66,2.88)

(8.17,4.44)

(4, -1) (6,1.81)

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Observations & Conclusions:

It was proposed that objective function to be minimized atnominal value without safety term

But observed to have no change in optimal solution in presentcase

Modication proposed is that the objective function alongwith the safety term should be minimized

This is due to objective being nonlinear in terms of designvariable

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Demonstrated case study with data perturbations only in objective function

q in inversely proportional to conservativeness of problem

If uncertainty in data is independent of others & strictly fol-lows corresponding intervals, the second modied method

gives better value with uncertainty satised

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Case Study 2: Three-bar truss problem

Objective: Minimize total structure weight

Constraints: Stress constraints on bar 1 and 2 (with bar 3identical to bar 1)36


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Problem Formulation:

Using linear elastic equations & Hookes law

2 E 40

(b1 + b3 ) ( b1 b3 )( b1 b3 ) ( b1 + b3 + 2 2 b2 )

z1z2

= P cos

P sin (17)

min b1 ,b2s.t.

f : 2 2 b1 + b2g1 :

22

P cos b1

+ P sin ( b1 + 2b2 ) 20 , 000 0

g2 : 2 P sin ( b1 + 2 b2 ) 20 , 000 0g3 : b1 0 , g4 : b2 0

(18)

Nominal load = 30,000 lb. & Angle = 45 o

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Nominal Optimal Solution:

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Data perturbations:

Uncertainty considered in angle of attack

0.45 % violation of rst stress constraint for 2.22 % pertur-bation in observed

Critical applications, cause structure to failRobust Formulation:

min b1 ,b2s.t.

f : 2 2 b1 + b2g1 + ( g1 ) q

20 , 000

0

g2 + ( g2 ) q 20 , 000 0g3 : b1 0 , g4 : b2 0

(19)

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Robust solution ( = 1 45):

Increase in weight = 24 %, uncertainty = 100 %

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Robust solution plot of stress in bar 1( = 1 45) :

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Robust solution plot of stress in bar 2( = 1 45) :

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Robust solution ( = 0.1 45) :

Increase in weight = 1.9 %, uncertainty = 10 %

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Robust solution plot of stress in bar 1( = 0.1 45) :

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Modication to methodology formulation:

min b1 ,b2

s.t.

f : 2 2 b1 + b2g1 + ( g1 ) q +

2

2 ( g1 ) 2 20 , 000 0g2 + ( g2 ) q +

2

2 ( g2 ) 2 20 , 000 0

g3 : b1 0 , g4 : b2 0

(20)

Max. deviation of constraint (rst-order robust) = L2 2 (L ismax. rate of change of sensitivity of gi )

For = 0.1 45 L = 19227, for = 1 45 L = 16866 &max. deviation from plot found less than L2

2

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Modication to methodology formulation:

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Modication to methodology formulation Plot for = 1 45 o:

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Further modication to methodology formulation L3 2 :

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Methodology provided good results in terms of objective cost

As increased, conservativeness increased

Deviation of actual stress from rst order approx. increasedwith

Methodology overestimated value of = too conservativewhen increases

Modied methodology L2 2 suitable for low values but notfor high values

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Second modication L3 2 can be approximately suitable forhigh values

Further investigation in the fractional value of L can be done

Methodology overestimates & underestimates the values of in various cases. Hence fractional value of L needs to have

balance for general applicability

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Summary & Conclusions:

Optimization, method to nd best possible value from a fea-sible set, has clear advantages over non optimal traditionaldesign methods

Data perturbations often occur in real world problems whichmake nominal optimal solution practically meaningless

Traditional methods to immunize solution were either limitedto soft constraints or were ultraconservative

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Robust linear optimization:

Methodology considered by Ben-Tal & Nemirovski was com-putationally tractable & with ellipsoidal uncertainty set (allworst values not at a time)

It had clear advantages over traditional interval model of un-certainty

Advantages of the model are visible only when uncertainty inone element. It was limited to inequality-only constraints

Both methods (interval & ellipsoidal) could be tried & thesolution depending on requirements could be chosen53


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Robust Nonlinear optimization:

Methodology considered by Y. Zhang used rst order approx.of constraints in neighborhood of nominal parameter to cal-culate optimal

General method can be applied to linear & nonlinear prob-lems

Can modify according to types of uncertainty

Can be applied to equality constraints as well54


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Cost of objective function low compared to uncertainty leveldesired

Addition of safety term to objective functions with data per-turbations effects the optimal solution.

Addition suggested to increase methods applicability to prob-lems with data perturbations only in objective

Breaking up uncertainty magnitude helps achieve good opti-mal value with objective being satised

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Method overestimates & underestimates uncertainty in differ-ent cases

Max. constraint violation capped by the rst-order robustness

Addition of second order term helps reduce constraint viola-tion

Lower fractional value of the added term suggested to havebalance between overestimated & underestimated case studies

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