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Mode Approach for a class of Mathematical Programs with Complementarity Constraints”. SIAM Journal on Optimization, 16, pp. 120–145. [56] Kannan, A., and Shanbhag, U., 2010. “Distributed iterative regularization algorithms for monotone nash games”. In Proceedings of IEEE Conference on Decision and Control (CDC). [57] Kannan, A., and Shanbhag, U., Forthcoming. “Distributed computation of equilibria in monotone nash games via iterative regularization techniques”. SIAM Journal on Optimization. [58] Jongen, H. T., Ruckmann, J. J., and Stein, O., 1997. “Disjunctive optimization: Critical point theory”. Journal of Optimization Theory and Applications, 93(2), May, pp. 321–336. [59] Gigerenzer, G., and Gaissmaier, W., 2011. “Heuristic decision making”. Annual Reviews in Psychology, 62, pp. 451– 82. A Several General Screening Rules Suppose there are R smooth screening “functions” σ i = (σ i,1 ,...,σ i,R ), σ i,r : X × R → R, from which screening rules can be derived. A (multiple) conjunctive screening rule requires the simultaneous satisfaction of one or more rules indexed by R = (r 1 ,...,r |R| ) ⊂ {1,...,R} and can be written as the vector inequality ⎡ ⎤ σ i,r1 (x, p) ⎢ ⎥ s i (x, p) = ⎣ . ⎦ ≤ 0; σ i,r|R| (x, p) equivalently, s i (x, p) = max r∈R {σ i,r (x, p)} ≤ 0. Disjunctions, on the other hand, represent the satisfaction of any individual rule from some set R ⊂ {1,...,R} and can be written s i (x, p) = min r {σ i,r (x, p)} ≤ 0; (see, e.g., [58]). Disjunctions of conjunctions are maximally general [30], and require the satisfaction of at least of C conjunctive rules each defined as the simultaneous satisfaction of the rules indexed by R c ⊂ {1,...,R}; i.e., s i (x, p) = min c { } max{σ i,r (x, p)} ≤ 0 r∈R c (see [58]). Subset conjunctive rules, a special case of disjunctions of conjunctions, require the satisfaction of any fixed number R ′ of a set of R screening rules, where R ′ ≤ R. While the exponential growth in the potential number of such rules threatens to make any model of non-compensatory decision processes intractable, there is evidence that people tend to use limited, simple rules in their decision-making [30, 59]. Consider vehicle choice as an example. Suppose x = (x 1 ,x 2 ,x 3 ) where x 1 is a numeric index for brand, x 2 is a numeric index for body style, and x 3 is a numeric index for powertrain type (Gasoline ICE, Diesel, Hybrid, etc). Let I = 1 (and suppress indexing with respect to i), R = 3, and suppose σ 1 (x, p) = −1 if x 1 is “Toyota”, and is +1 otherwise; σ 2 (x, p) = −1 if x 2 is “sedan”, and is +1 otherwise; σ 3 (x, p) = −1 if x 3 is “Hybrid”, and is +1 otherwise. The following illustrate the categories of rules discussed above: The conjunctive rule “I’ll only consider hybrids” is σ 3 (x, p) ≤ 0. The negated conjunctive rule “I won’t consider hybrids” is −σ 3 (x, p) ≤ 0. The conjunctive rule “I’ll consider any Toyota hybrid” is s(x, p) = [ ] σ1 (x, p) ≤ 0. σ 3 (x, p) The disjunctive rule “I’ll consider any Toyota or any hybrid” is s(x, p) = min{σ 1 (x, p),σ 3 (x, p)} ≤ 0. The subset conjunctive rule “I’ll consider a Toyota sedan, a Toyota hybrid, a hybrid sedan, or a Toyota hybrid sedan” is { s(x, p) = min max{σ 1 (x, p),σ 2 (x, p)} , max{σ 2 (x, p),σ 3 (x, p)} , } max{σ 1 (x, p),σ 3 (x, p)} ≤ 0. The general disjunction-of-conjunctions rule “I’ll consider a Toyota hybrid or a sedan” is { s(x, p) = min max{σ 1 (x, p),σ 3 (x, p)} , σ 2 (x, p) } ≤ 0. B VEHICLE DESIGN AND CHOICE MODELS Fuel consumption (in gpm) and 0-60 acceleration time (in s) are related by g(a) = 0.035 + 53.5 + 69.5e−a − 1.8a 1.4 + 106.9/a 1000 (24) 16 Copyright c○ 2012 by ASME
This formula is a modified version of the model estimated by [39], and plotted in Fig. 1. Our modifications are meant to ensure that acceleration performance and fuel economy are strongly inversely related: very fast acceleration times can be obtained with very low fuel economy, and very slow acceleration times are required for very high fuel economies. Unit costs are also a function of acceleration/fuel economy performance, given by ( c(a) = e a/12 1.5 + 1.97e −a − 0.04a + 1 ) . (25) a − 1.5 See Fig. 2. This formula is also modeled after the estimates in [39]. C PROOFS Proof. Proof of Lemma 1: We prove (i), with (ii) and (iii) easy corollaries. Suppose that there are two distinct vectors ω,ω ′ ∈ Ω that are both strictly choice consistent with (X,p). Then for some (i, j,r), ω i, j,r ≠ ω i, ′ j,r . But by strict choice consistency, ω i, j,r = 1 = ω i, ′ j,r if s i,r(x j , p j ) ≤ 0 and ω i, j,r = 0 = ω i, ′ j,r if s i,r(x j , p j ) > 0. This contradiction proves that ω = ω ′ . □ Proof. Proof of Lemma 2: Let (ω,X,p) be feasible, and consider any (i, j,r). Suppose s i,r (x j , p j ) > 0. Then because ω i, j,r s i,r (x j , p j ) ≤ 0, ω i, j,r = 0. On the other hand, suppose s i,r (x j , p j ) < 0. Then because (1 − ω i, j,r )s i,r (x j , p j ) ≥ 0, ω i, j,r = 1. If s i,r (x j , p j ) = 0, then ω i, j,r can be either 0 or 1. Because one of these conditions must hold for every (i, j,r), ω is almost choice-consistent. □ Proof. Proof of Lemma 3: By Lemma 1(ii), π C (X,p) = ˆπ( ¯ω(X,p),X,p). Thus, because ( ¯ω(X,p),X,p) is feasible for Eqn. (15) for any feasible (X,p), the optimal value π C,∗ can be attained in Eqn. (15) by some feasible (ω,X,p). Thus, the optimal value of Eqn. (15) must be at least π C,∗ . For the second claim, assume that (ω,X,p) is a local solution of Eqn. (15) and ω is choice-consistent with (X,p). By Lemma 1, ω = ¯ω(X,p) and π C (X,p) = ˆπ R (ω,X,p). Local optimality means that ˆπ R (ω,X,p) ≥ ˆπ R (ω ′ ,X ′ ,p ′ ) for all (ω ′ ,X ′ ,p ′ ) in a (feasible) neighborhood U of (ω,X,p). Let D be the projection of U onto the X and p coordinates. Because ( ¯ω(X ′ ,p ′ ),X ′ ,p ′ ) ∈ U for all (X ′ ,p ′ ) ∈ D, π C (X,p) = ˆπ R (ω,X,p) ≥ ˆπ( ¯ω(X ′ ,p ′ ),X ′ ,p ′ ) = π C (X ′ ,p ′ ). Thus (X,p) is locally optimal for Eqn. (9). □ Proof. Proof of Lemma 4: The proof given for Lemma 2 applies essentially unchanged. □ Proof. Proof of Lemma 5: The proof given for Lemma 3 applies essentially unchanged. □ D TRIVIAL KKT POINTS IN MPCC RELAXATIONS An additional complication makes solution of Eqn. (15) or (16) potentially difficult: the KKT conditions have “trivial” solutions that are not necessarily profit-optimal. Standard NLP methods may be attracted to the spurious solutions represented by these points, rather than to actual solutions. We first prove that there are trivial KKT points in the example from Section 3, and then demonstrate the problem with numerical results. The MPCC relaxation for Eqn. (6) is maximize ˆπ(ω,a, p) with respect to 0 ≤ ω ≤ 1, 2.5 ≤ a ≤ 15, p ≥ 0 subject to − ω(k(a, p) − B) ≥ 0 (1 − ω)(k(a, p) − B) ≥ 0 The KKT conditions for Eqn. (26) are the MCP 0 ≤ ω ≤ 1 ⊥ −(D w ˆπ)(ω,a, p) + (λ 1 + λ 2 )(k(a, p) − B) (26) 2.5 ≤ a ≤ 15 ⊥ −(D a ˆπ)(ω,a, p) + (ω(λ 1 + λ 2 ) − λ 2 )(D a k)(a, p) 0 ≤ p ⊥ −(D p ˆπ)(ω,a, p) + (ω(λ 1 + λ 2 ) − λ 2 )(D p k)(a, p) ≥ 0 0 ≤ λ 1 ⊥ −ω(k(a, p) − B) ≥ 0 0 ≤ λ 2 ⊥ (1 − ω)(k(a, p) − B) ≥ 0 (27) This “MCP form” of the KKT conditions follows from the standard KKT conditions [35] and the definition of an MCP [47–49]; see Appendix E. The existence of trivial solutions to these equations is a consequence of the following result: Lemma 6. (D a ˆπ)(0,a, p) = 0, (D p ˆπ)(0,a, p) = 0, and (D w ˆπ)(0,a, p) ≥ 0 for all feasible a and p ≥ c(a). Proof. The specific claims are a consequence of the following general formulae: □ (D w ˆπ)(w,a, p) = eu(a,p) 1 + we u(a,p) (1 − PC (w,a, p))(p − c(a)), (D a ˆπ)(w,a, p) = (D a u)(a, p)P C (w,a, p)(1 − P C (w,a, p))(p − c(a)) (D p ˆπ)(w,a, p) − (D a c)(a)P C (w,a, p), = (D p u)(a, p)P C (w,a, p)(1 − P C (w, p,e))(p − c(a)) + P C (w,a, p). Corollary 1. For any feasible a and λ ≥ 0, there exists some ¯p(a,λ) > 0 such that the KKT conditions for Eqn. (26), Eqn. (27), are solved by (0,a, p,λ,0) for any p ≥ ¯p(a). 17 Copyright c○ 2012 by ASME
Page 1 and 2: Proceedings of the ASME 2012 Intern
Page 3 and 4: j ∈ {1,...,J} is given by ⎧ ⎪
Page 5 and 6: TABLE 1. Optimal solutions to Eqns.
Page 7 and 8: over the assumed collection of cons
Page 9 and 10: y either relaxing the constraints c
Page 11 and 12: also has different compensatory pre
Page 13 and 14: Vehicle Fuel Economy (mpg) 50 45 40
Page 15: pp. 366-387. [27] Hauser, J. R., Fo

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