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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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Major Advances in Robust Optimization
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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Major Advances in Robust Optimization
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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Observations & Conclusions:
Demonstrated case study with data perturbations only in ob- jective 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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Observations & Conclusions:
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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Observations & Conclusions:
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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Summary & Conclusions:
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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Summary & Conclusions:
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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Summary & Conclusions:
Robust Nonlinear optimization:
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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Summary & Conclusions:
Robust Nonlinear optimization:
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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