Environmental contours and optimal design
EEnvironmental contours and optimal design
September 9, 2020
Kristina Rognlien Dahl Department of Mathematics, University of Oslo,Norway ([email protected]). This work has received funding from the Nor-wegian Research Council SCROLLER project, project number 299897.Arne Bang Huseby Department of Mathematics, University of Oslo, Norway([email protected])
Abstract
Classical environmental contours are used in structural design in or-der to obtain upper bounds on the failure probabilities of a large classof designs. Buffered environmental contours, first introduced in [4], servethe same purpose, but with respect to the so-called buffered failure prob-ability. In contrast to classical environmental contours, buffered environ-mental contours do not just take into account failure vs. functioning, butalso to which extent the system is failing. This is important to take intoaccount whenever the consequences of failure are relevant. For instance,if we consider a power network, it is important to know not just that thepower supply is failed, but how many consumers are affected by the failure.In this paper, we study the connections between environmental contours,both classical and buffered, and optimal structural design. We connectthe classical environmental contours to the risk measure value-at-risk.Similarly, the buffered environmental contours are naturally connected tothe convex risk measure conditional value-at-risk. We study the problemof minimizing the risk of the cost of building a particular design. Thisproblem is studied both for value-at-risk and conditional-value-at-risk. Byusing the connection between value-at-risk and the classical environmentalcontours, we derive a representation of the design optimization problemexpressed via the environmental contour. A similar representation is de-rived by using the connection between conditional value-at-risk and thebuffered environmental contour. From these representations, we derive asufficient condition which must hold for an optimal design. This is doneboth in the classical and the buffered case. Finally, we apply these resultsto solve a design optimization problem from structural reliability.
Key words:
Structural reliability analysis, environmental contour, struc-tural design, failure probability, buffered failure probability, design optimization.1 a r X i v : . [ m a t h . O C ] S e p Introduction
In this paper, we will consider the problem of design optimization. We willminimize the risk of the cost of a structural design. The cost of the structuraldesign is composed of two parts: A fixed failure cost K which occurs in case ofsystem failure and a cost function κ ( x ) which only depends on the chosen design x . This risk-of-cost minimization will be done with respect to two different riskmeasures: Value-at-risk and conditional value-at-risk. Conditional value-at-riskis a convex risk measure, which takes into account not just whether a systemfunctions or fails, but to which extent it fails. We connect the value-at-riskand conditional value-at-risk to environmental contours via functions C ( u ) and ¯ C ( u ) . This connection to the functions C ( u ) and ¯ C ( u ) allows us to get analternative characterisation of the risk-minimization problems.The structure of the paper is as follows: In Section 2, we recall the definitionof value-at-risk and derive som properties of this risk measure. In Section 3,we recall the concept of environmental contours and buffered environmentalcontours. In Section 4, we derive an alternative characterization of the designoptimization problem of minimizing the value-at-risk of the cost of a structureby connecting this problem to environmental contours. In Section 5, we applythis methodology to a structural design problem. Value-at-risk ignores the tailof the distribution of the structure function, therefore we recall the definitionof another risk measure, conditional value-at-risk (CVaR), in Section 6. Wealso derive some properties of CVaR. In Section 7, we minimize the conditionalvalue-at-risk of the cost of a structure. We derive an alternative characterizationof this problem by connecting it to buffered environmental contours. Finally,in Section 8, we discuss a criterion for selecting a set of initial design conceptsrelated to the system of interest. Let X be a random variable, representing risk. Define S X ( x ) := P ( X > x ) . Let α ∈ (0 , be a given probability representing an acceptable level of risk . In thecontext of structural design, the value of α can for instance be determined bythe firm based on the required return period of the system. See [4] for furtherdetails. The α -level value-at-risk associated with the risk X , denoted by V α [ X ] ,is given by S − X ( α ) . More formally, we define:(2.1) V α [ X ] = S − X ( α ) = inf { x : P ( X > x ) ≤ α } . value-at-risk is frequently used for risk management in banks and insurancecompanies. For more about value-at-risk as a risk measure, see e.g. [9] and [2].In the special case where X is absolutely continuously distributed, we have: V α [ X ] = S − X ( α ) = x if and only if P ( X > x ) = α. S X is strictly decreasing, we have that:(2.2) V α [ X ] = x if and only if P ( X > x ) ≤ α ≤ P ( X ≥ x ) . Finally, if X is a discrete random variable , we have that:(2.3) V α [ X ] = x if and only if P ( X > x ) ≤ α < P ( X ≥ x ) . We now show some properties of value-at-risk which are needed to derive analternative characterization of our design optimization problem in Section 4.
Theorem 2.1 (Monotone transform)
For any strictly increasing continu-ous function φ : R → R we have: (2.4) V α [ φ ( X )] = S − φ ( X ) ( α ) = φ ( S − X ( α )) Proof:
We note that since φ is strictly increasing, it follows by (2.1) that: V α [ φ ( X )] = inf { y : P ( φ ( X ) > y ) ≤ α } (2.5) = inf { y : P ( X > φ − ( y )) ≤ α } . (2.6)We then substitute y = φ ( x ) and φ − ( y ) = x , and get: V α [ φ ( X )] = inf { φ ( x ) : P ( X > x ) ≤ α } (2.7) = φ (inf { x : P ( X > x ) ≤ α } ) (2.8) = φ ( S − X ( α )) . (2.9) (cid:3) Value-at-risk is linear, as shown in the following result.
Corollary 2.2 (Linearity)
For a > and b ∈ R we have: V α [ aX + b ] = aV α [ X ] + b. Proof:
The result follows directly from the monotonicity property by notingthat: φ ( X ) = aX + b is a strictly increasing function for all a > and b ∈ R . (cid:3) Environmental contours The classical approach to environmental contours was first introduced in[13]. A Monte Carlo approach to environmental contours was considered in [6],[7] and [8].In probabilistic structural design, it is common to define a performance func-tion g ( x , V ) depending on some deterministic design variables x = ( x , x , . . . , x m ) (cid:48) representing various parameters such as capacity, thickness, strength etc. andsome random environmental quantities V = ( V , V , . . . , V n ) (cid:48) ∈ V , where V ⊆ R n . The performance function is defined such that if g ( x , V ) > , the structureis failed , while if g ( x , V ) ≤ , the structure is functioning . Moreover, for agiven x the set F = { v ∈ V : g ( x , V ) > } is called the failure region of thestructure. An important part of the probabilistic design process is to make surethat P ( V ∈ F ) is acceptable for all failure regions F of interest, denoted E .In order to avoid failure regions with unacceptable probabilities, it is neces-sary to put some restrictions on the family of failure regions. This is done byintroducing a set B ⊆ R n chosen so that for any relevant failure region F whichdo not overlap with B , the failure probability P ( V ∈ F ) is small . The family E is chosen relative to B so that F ∩ B ⊆ ∂ B for all F ∈ E , where ∂ B denotes theboundary of B . This boundary is then referred to as an environmental contour .See Figure 3.1. V V B ∂ BF Environmental contourFailure region
Figure 3.1: An environmental contour ∂ B and a failure region F .Following [5] we define the exceedence probability of B with respect to E as:(3.1) P e ( B , E ) := sup { p f ( F ) : F ∈ E} . For a given target probability α the objective is to choose an environmentalcontour ∂ B such that: P e ( B , E ) = α This section is based on [4]. We include it here for the sake of completeness. E . Of particular interest are cases where one can argue that the failureregion of a structure is convex . That is, cases where E is the class of all convexsets which do not intersect with the interior of B . There are many possible ways of constructing environmental contours. In thispaper we connect the design optimization problem to the
Monte Carlo basedapproach to environmental contours , first introduced in [6], and improved in [7]and [8].Let U be the set of all unit vectors in R n , and let u ∈ U . We then introducea function C ( u ) defined for all u ∈ U as:(3.2) C ( u ) := inf { C : P ( u (cid:48) V > C ) ≤ α } Thus, C ( u ) is the (1 − α ) -quantile of the distribution of u (cid:48) V . Given the distribu-tion of V , the function C ( u ) can be estimated by using Monte Carlo simulation,see e.g. [4].Then, from the previous definitions, P [ u (cid:48) V > C ( u )] = α. We will use this equality to connect the optimal design problem to environmentalcontours, via the quantile function C ( u ) . Similarly, so called buffered environmental contours , first introduced in [4], canbe estimated via a function(3.3) ¯ C ( u ) := E [ u (cid:48) V | u (cid:48) V > C ( u )] . Buffered environmental contours are constructed similarly to classical envi-ronmental contours, with the exception that the failure probability of interest isthe buffered failure probability . For any probability level α , the α - superquantile of g ( x , V ) , ¯ q α ( x ) , is defined as:(3.4) ¯ q α ( x ) = E [ g ( x , V ) | g ( x , V ) > q α ( x )] . That is, the α -superquantile is the conditional expectation of g ( x , V ) when weknow that its value is greater than or equal the α -quantile. Then, the bufferedfailure probability, ¯ p f , first introduced by Rockafellar and Royset [11], is definedas(3.5) ¯ p f ( x ) = 1 − α, α is chosen so that ¯ q α ( x ) = 0 From these definitions, it follows that(3.6) ¯ p f ( x ) = P ( g ( x , V ) > q α ( x )) = 1 − F ( q α ( x )) where F denotes the distribution of the structure function g . Buffered envi-ronmental contours can be constructed via Monte Carlo similarly as classicalcontours. We will connect the design optimization problem wrt. conditionalvalue-at-risk to buffered environmental contours in Section 7. We will now connect the optimal design problem with respect to value-at-risk tothe quantile function C ( u ) . Then, we use this connection to derive an alternativecharacterization of the optimization problem. Some key references on designoptimization and structural design are [10] and [3].Let V = ( V , . . . , V n ) ∈ V be a vector of environmental variables and let α ∈ (0 , be a given probability representing an acceptable level of risk. Weassume that we have determined a function C ( u ) defined for all unit vectors u ∈ R n such that:(4.1) P [ u (cid:48) V > C ( u )] = α, for all u ∈ R n . We also introduce the following notation: Π( u ) = { V ∈ V : u (cid:48) V = C ( u ) } , Π + ( u ) = { V ∈ V : u (cid:48) V > C ( u ) } , Π − ( u ) = { V ∈ V : u (cid:48) V ≤ C ( u ) } Hence, we have:(4.2) P [ V ∈ Π + ( u )] = P [ u (cid:48) V > C ( u )] = α, for all u ∈ R n . Remark 4.1 (Connection to MC contours)
Note that this is the same frame-work as what is frequently used in connection to Monte Carlo environmental con-tours, see Section 3 as well as [4]. The function C corresponds to the quantilefunction used to construct environmental contours, see (3.2) . Let the cost of system failure be denoted by K . We introduce a deterministicfunction κ = κ ( x ) representing the cost of the design x , and assume that: κ ( x ) < K for all x ∈ X . Note that this assumption implies that for any design of interest, system failurecosts more than rebuilding the system. This means that system failure has6ther financial consequences than just having to rebuild the system. This willtypically be the case in practise, for instance for telecommunication networks,subway networks or power production companies.The total cost , denoted H , is given by: H ( V , x ) = K · I [ g ( V , x ) >
0] + κ ( x ) . where I [ · ] denotes the indicator function. The α -level value-at-risk of a givendesign, denoted V α ( H ) , is given by: V α ( H ) = S − H ( α ) , where S H ( h ) = 1 − F H ( h ) = P ( H > h ) . Thus, V α ( H ) is the (1 − α ) -percentileof the distribution of H .Our main objective is to choose a design x so that to minimize the value-at-risk of H , i.e. min x ∈X V α (cid:0) H ( V , x ) (cid:1) Since κ ( x ) is deterministic, it follows by the linearity of V α that: V α [ H ] = V α [ K · I [ g ( V , x ) > κ ( x ) . We observe that K · I [ g ( V , x ) > is a discrete random variable with onlytwo possible values, and K . Its distribution is given by: P [ K · I [ g ( V , x ) >
0] = K ] = P [ g ( V , x ) > ,P [ K · I [ g ( V , x ) >
0] = 0] = P [ g ( V , x ) ≤ . By (2.3) we know that: V α [ K · I [ g ( V , x ) > y, if and only if: P [ K · I [ g ( V , x ) > > y ] ≤ α< P [ K · I [ g ( V , x ) > ≥ y ] In particular, we have P [ K · I [ g ( V , x ) > > K ] = 0 < α. This implies that: V α [ K · I [ g ( V , x ) > K, if and only if: P [ K · I [ g ( V , x ) > ≥ K ] = P [ g ( V , x ) > > α Furthermore, we have P [ K · I [ g ( V , x ) > ≥
0] = 1 > α. V α [ K · I [ g ( V , x ) > , if and only if: P [ K · I [ g ( V , x ) > >
0] = P [ g ( V , x ) > ≤ α. Summarizing this, we get:(4.3) V α ( K · I [ g ( V , x ) > (cid:40) K if P [ g ( V , x ) > > α if P [ g ( V , x ) > ≤ α From this it follows that:(4.4) V α ( H ) = (cid:40) K + κ ( x ) if P [ g ( V , x ) > > ακ ( x ) if P [ g ( V , x ) > ≤ α Since we have assumed that κ ( x ) < K for all x ∈ X , it follows that an optimaldesign x must be chosen so that:(4.5) P [ g ( V , x ) > ≤ α Theorem 4.2 (Halfspace condition VaR)
A sufficient condition for (4.5) to hold is that g ( V , x ) ≤ for all V such that u (cid:48) V ≤ C ( u ) , where u ∈ R n is asuitably chosen unit vector. Proof:
The condition implies that if g ( V , x ) > , then u (cid:48) V > C ( u ) . Hence,by (4.1) we get that: P [ g ( V , x ) > ≤ P [ u (cid:48) V > C ( u )] = α. Therefore, we conclude that inequality (4.5) is satisfied. (cid:3)
We then let u ∈ R n be a unit vector and consider the following subclass ofdesigns: X ( u ) = { x ∈ X : g ( V , x ) ≤ for all V ∈ Π − ( u ) } . i.e., designs such that the systems functions for all V ∈ Π − ( u ) . By the halfspacecondition, Theorem 4.2, we know that condition (4.5) is satisfied for all designs x ∈ X ( u ) . Hence, an optimal design within the subclass X ( u ) can be found byminimising κ ( x ) with respect to x ∈ X ( u ) . Different choices of the unit vector u will generate different optimal designs. However, the choice of u may oftenbe a result of initial concept decisions related to the system of interest. Thus,it may not be necessary to consider multiple subclasses of design.8 Example: Structural reliability
We consider a system whose performance depends on the non-negative environ-mental variables, V = ( V , . . . , V n ) ∈ V . The system fails if: A V > x where A = A m × n is a matrix, and the design x = ( x , . . . , x m ) is a vector of strengths .The cost of the design x is given by: κ ( x ) = c x + · · · + c m x m . We want to minimize κ ( x ) subject to P [ A V > x ] ≤ α. Since this failure probability may be difficult to compute, we instead mini-mize κ ( x ) subject to:(5.1) { V ∈ V : A V > x } ⊆ { V ∈ V : u (cid:48) V > C ( u ) } . It follows that if the design x satisfies (5.1), then: P [ A V > x ] ≤ P [ u (cid:48) V > C ( u )] = α. For a given design x , we can then check if it satisfies condition (5.1) bysolving the following LP-problem:(5.2) Minimise u (cid:48) V subject to A V ≥ x .Let V denote the solution to the minimization problem (5.2). Then x satisfies condition (5.1) if and only if: u (cid:48) V > C ( u ) . By using a suitable iteration method one can then find a design x which mini-mizes κ ( x ) subject to condition (5.1). So far, we have used value-at-risk as a design criterion. The problem with thisis that VaR ignores the size of the outcomes in the tail of the distribution.
Example 6.1 (Value-at-risk ignores the tail)
VaR . (X) is the x − valuesuch that only of the outcomes of X are larger (i.e., worse in our context)than this value. Hence, VaR . (X) ignores the size, and hence the consequences,of all values above this level. C α , is defined as(6.1) C α ( X ) := 1 α (cid:90) α V u ( X ) du That is, we compute the average of the value-at-risk in the α % worst cases.Coherent risk measures, which conditional value-at-risk is an example of, werefirst introduced in [1]. CVaR is frequently used in mathematical finance, andto some extent in insurance mathematics. [12] and [14] study optimizationtechniques in connection to CVaR.Note that CVaR is also a convex risk measure, i.e. ( i ) (Convexity) For ≤ λ ≤ , C α ( λX + (1 − λ ) Y ) ≤ λC α ( X ) + (1 − λ ) C α ( Y ) . ( ii ) (Monotonicity) If X ≥ Y , then C α ( X ) ≥ C α ( Y ) . ( iii ) (Translation invariance) If m ∈ R , then C α ( X + m ) = C α ( X ) − m . Remark 6.2
The monotonicity property is the other way around from what iscommon in financial mathematics because we view large positive values as bad(failure of system). In finance, greatly negative values are bad (losses).
Theorem 6.3
For any strictly increasing continuous function φ : R → R wehave: (6.2) C α [ φ ( X )] = 1 α (cid:90) α φ ( V u ( X )) du. Proof:
From the definition of CVaR (6.1): C α [ φ ( X )] = 1 α (cid:90) α S − φ ( X ) ( u ) du = 1 α (cid:90) α φ ( S − X ( u )) du = 1 α (cid:90) α φ ( V u ( X )) du (cid:3) In order to prove a monotone transform property of conditional value-at-risk, we need the following well-known inequality, included here for the sake ofcompleteness:
Theorem 6.4 (Jensen’s inequality)
Let (Ω , F , P ) be a probability space. Let g : Ω → R be a P -integrable function. Also, assume that ϕ : R → R is a convexfunction. Then, ϕ ( (cid:90) Ω g ( ω ) dP ( ω )) ≤ (cid:90) Ω ϕ ( g ( ω )) dP ( ω ) . From Jensen’s inequality, we find that for f : [ a, b ] → R , ϕ : R → R convex,we have ϕ (cid:0) b − a (cid:90) ba f ( x ) dx (cid:1) ≤ b − a (cid:90) ba ϕ ( f ( x )) dx. By using this, we can prove the following monotone transform property ofCVaR:
Theorem 6.5 (Monotone transform of CVaR)
Assume that φ : R → R isa strictly increasing, continuous and convex function. Then, (6.3) φ ( C α [ X ]) ≤ C α [ φ ( X )] . Proof: φ ( C α [ X ]) = φ ( α (cid:82) α S − X ( u )) ≤ α (cid:82) α φ ( S − X ( u )) du = α (cid:82) α S − φ ( X ) ( u ) du = C α ( φ ( X )) . Here, the inequality holds from Jensen’s inequality. The second to last equalityfollows because of equation (2.4) of the monotone transform proposition forVaR. (cid:3)
Conditional value-at-risk is linear, as shown in the following result:
Corollary 6.6 (Linearity of CVaR)
For a > and b ∈ R we have: C α [ aX + b ] = aC α [ X ] + b. roof: By using the definition of CVaR and the linearity of VaR, we see that C α ( aX + b ) = α (cid:82) α V u ( aX + b ) du = α (cid:82) α { aV u ( X ) + b } du = a ( α (cid:82) α V u ( X ) du ) + b = aC α ( X ) + b. (cid:3) Parallel to the VaR-case, we would like to choose an optimal design x such thatthe conditional value at risk of the total cost is minimized: min x ∈X C α ( H ( V , x )) where, as before, H ( V , x ) = K · I [ g ( V , x ) >
0] + κ ( x ) . From the linearity ofCVaR (see Corollary 6.6), C α ( H ) = K · C α ( I [ g ( V , x ) > κ ( x ) . Note that V u is decreasing in u from its definition. Also, note that(7.1) C α ( I [ g ( V , x ) > α (cid:82) α V u ( I [ g ( V , x ) > du = α (cid:82) min { P ( g ( V , x ) > ,α } du = α min { P ( g ( V, x ) > , α }≥ V α ( I [ g ( V , x ) > . Here, the second equality follows from (4.3)-(4.4). The inequality follows fromthe formula for V α ( I [ g ( V , x ) > in equation (4.3). Also, if P ( g ( V , x ) > >α , we see that(7.2) min { P ( g ( V , x ) > , α } = α. Hence C α = 1 (the same as V α in this case). The property in (7.1) is also true ingeneral: Conditional value-at-risk, C α , is more conservative than value-at-risk, V α .Now, consider two cases: Let case be the case where P [ g ( V , x ) > > α, and case be the case where P [ g ( V , x ) > ≤ α. C α ( H ) = (cid:40) K + κ ( x ) in case K P [ g ( V , x ) > α + κ ( x ) in case . Note that ≤ P [ g ( V , x ) > α ≤ in case 2 above (since P [ g ( V , x ) > ≤ α ). Also,note that our assumption that κ ( x ) < K for all x ∈ X , is no longer enough toguarantee that the optimal design should be chosen such that P [ g ( V , x ) > ≤ α . By considering the difference between the two cases in equation (7.3), we findthat a sufficient condition to ensure that the optimal design satisfies P [ g ( V , x ) > ≤ α is:(7.4) κ ( x ) − κ ( x ) ≤ Kα ( α − P [ g ( V , x ) > for all x such that P [ g ( V , x ) > ≤ α and x (that is, case ) such that P [ g ( V , x ) > > α (i.e., case ). Note that this slightly resembles a Lipschitzcondition for the cost function κ ( · ) .Assume, like before, that we have determined a function C ( u ) defined forall unit vectors u ∈ R n such that (3.2) holds. Now, define a function, ¯ C ( u ) , asfollows(7.5) ¯ C ( u ) := E [ u (cid:48) V | u (cid:48) V > C ( u )] . Furthermore, introduce the following notation: ¯Π( u ) = { V ∈ V : u (cid:48) V = ¯ C ( u ) } , ¯Π + ( u ) = { V ∈ V : u (cid:48) V > ¯ C ( u ) } , ¯Π − ( u ) = { V ∈ V : u (cid:48) V ≤ ¯ C ( u ) } and define(7.6) Γ( u , V ) := u · V − ¯ C ( u ) . Remark 7.1 (Connection to buffered contours)
Note that this is the sameframework as what is used in connection to buffered environmental contours, see[4]. The function ¯ C corresponds to the superquantile function used to constructbuffered environmental contours, see (3.2) . For a fixed (but arbitrary) unit vector u , let ¯ X ( u ) denote the set of designs x such that g ( · , x ) dominated by Γ( u , · ) . Then, for any x ∈ ¯ X ( u ) , P ( g ( V , x ) > ≤ P (Γ( u , x ) > P ( u · V − ¯ C ( u ) > P ( u · V > ¯ C ( u ))= P ( ¯Π + ( u )) ≤ P (Π + ( u ))= α. ¯ C ( u ) > C ( u ) , so by the definitions of ¯Π + ( u ) and Π + ( u ) , we find that ¯Π + ( u ) ⊆ Π + ( u ) . Hence, P ( ¯Π + ( u )) ≤ P (Π + ( u )) . Therefore, we have proved that if x ∈ ¯ X ( u ) , then(7.7) P ( g ( V , x ) > ≤ α. We summarize this in the following theorem.
Theorem 7.2 (Domination condition CVaR)
A sufficient condition for (7.7) to hold is that g ( · , x ) is dominated by a function Γ( u , · ) of the form (7.6) , where u ∈ R n is a suitably chosen unit vector. If condition (7.4) is satisfied, we know that the optimal design should bechosen such that equation (7.7) holds. Let u ∈ R n be a suitably chosen unitvector. By the domination condition for CVaR, Theorem 7.2, we know that thecondition (7.7) is satisfied for all designs x ∈ ¯ X ( u ) . Hence, an optimal designis found by minimising K P [ g ( V , x ) > α + κ ( x ) with respect to x ∈ ¯ X ( u ) . u Different choices of the unit vector u will generate different optimal designs.The choice of u may often be a result of initial concept decisions related to thesystem. If a firm has N different initial concepts, u , . . . , u N under considera-tion, the minimization problem can be solved for each of these u i ’s, i = 1 , . . . , N .This results in N potentially optimal designs x , . . . , x N .To find the optimal concept, the firm can compare the objective functionvalues, i.e. V α ( H ( V , x i )) or C α ( H ( V , x i )) , i = 1 , . . . , N , of these designs.Assume that for a fixed design x , we know that the corresponding performancefunction g ( · , x ) is monotone in some V i -component, i = 1 , . . . , n . Then oneshould choose the unit vector u such that it "follows the monotonicity". That is,if g is non-decreasing in V i , so V i ≤ ¯ V i implies that g (( V i , V ) , x ) ≤ g (( ¯ V i , V ) , x ) ,then u should be chosen such that u i ∈ (0 , .If g is non-increasing in V i , so V i ≤ ¯ V i implies that g (( V i , V ) , x ) ≥ g (( ¯ V i , V ) , x ) ,then u should be chosen such that u i ∈ ( − , .We make the previous statement more precise: Consider the VaR case. TheCVaR case is parallel. Assume that there exists V , V i ≤ ¯ V i where the systemfails in ( ¯ V i , V ) , but functions in ( V i , V ) . Note that this assumption is slightlystricter than g being monotone in component i . It corresponds to monotonicity The notation ( V i , V ) means the vecor V where component i is V i
14s well as criticality of the i ’th environmental component. Also, assume forcontradiction that u i ∈ ( − , .By assumption, g (cid:0) ( V i , V ) , x (cid:1) ≤ and g (cid:0) ( ¯ V i , V ) , x (cid:1) > . That is, the system fails in ( ¯ V i , V ) , but functions in ( V i , V ) . There existsa vector u such that the (by scaling) unit vector ( u i , u ) satisfies ( ¯ V i , V ) ∈ Π − (( u i , u )) and ( V i , V ) ∈ Π + (( u i , u )) .From the definitions of Π + (( u i , u )) and Π − (( u i , u )) , this implies that thesystem should function in ( ¯ V i , V ) and fail in ( ¯ V i , V ) . But this contradicts theassumption. Hence, choosing u i ∈ ( − , leads to a contradiction, so u i shouldbe chosen in the only other way possible, namely such that u i ∈ (0 , . Thearguments in the case where g is non-increasing in V i is parallel. So far, we have minimized the risk of the cost of a structural design wrt. VaRand CVaR.An alternative design optimization problem is to minimize the expected costunder a risk constraint:(9.1) min E [ H ( x , V )] such that risk ( g ( x , V )) ≤ α. Here, the risk-function, which depends on the performance function of thesystem, can be either value-at-risk or conditional value-at-risk.By looking at this optimization problem, the environmental contour becomesa representation of the constraint. This problem and its connection to environ-mental contours are the topic of future works.
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