Discrete Choice Prox-Functions on the Simplex

We derive new prox-functions on the simplex from additive random utility models of discrete choice. They are convex conjugates of the corresponding surplus functions. In particular, we explicitly derive the convexity parameter of discrete choice prox-functions associated with generalized extreme value models, and specifically with generalized nested logit models. Incorporated into subgradient schemes, discrete choice prox-functions lead to a probabilistic interpretations of the iteration steps. As illustration, we discuss an economic application of discrete choice prox-functions in consumer theory. The dual averaging scheme from convex programming adjusts demand within a consumption cycle.

Rates of superlinear convergence for classical quasi-Newton methods

We study the local convergence of classical quasi-Newton methods for nonlinear optimization. Although it was well established a long time ago that asymptotically these methods converge superlinearly, the corresponding rates of convergence still remain unknown. In this paper, we address this problem. We obtain first explicit non-asymptotic rates of superlinear convergence for the standard quasi-Newton methods, which are based on the updating formulas from the convex Broyden class. In particular, for the well-known DFP and BFGS methods, we obtain the rates of the form $$(\frac{n L^2}{\mu ^2 k})^{k/2}$$ ( n L 2 μ 2 k ) k / 2 and $$(\frac{n L}{\mu k})^{k/2}$$ ( nL μ k ) k / 2 respectively, where k is the iteration counter, n is the dimension of the problem, $$\mu $$ μ is the strong convexity parameter, and L is the Lipschitz constant of the gradient.

Locally Convex Words and Permutations

We introduce some new classes of words and permutations characterized by the second difference condition $\pi(i-1) + \pi(i+1) - 2\pi(i) \leq k$, which we call the $k$-convexity condition. We demonstrate that for any sized alphabet and convexity parameter $k$, we may find a generating function which counts $k$-convex words of length $n$. We also determine a formula for the number of 0-convex words on any fixed-size alphabet for sufficiently large $n$ by exhibiting a connection to integer partitions. For permutations, we give an explicit solution in the case $k = 0$ and show that the number of 1-convex and 2-convex permutations of length $n$ are $\Theta(C_1^n)$ and $\Theta(C_2^n)$, respectively, and use the transfer matrix method to give tight bounds on the constants $C_1$ and $C_2$. We also providing generating functions similar to the the continued fraction generating functions studied by Odlyzko and Wilf in the "coins in a fountain" problem.

Multivariate Distributions with Fixed Marginals and Correlations

Consider the problem of drawing random variates (X1, …, Xn) from a distribution where the marginal of each Xi is specified, as well as the correlation between every pair Xi and Xj. For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between Xi and Xj. Any achievable correlation between Xi and Xj is a convex combination of these bounds. We call the value λ(Xi, Xj) ∈ [0, 1] of this convex combination the convexity parameter of (Xi, Xj) with λ(Xi, Xj) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F1, …, Fn of (X1, …, Xn), we show that λ(Xi, Xj) = λij if and only if there exist symmetric Bernoulli random variables (B1, …, Bn) (that is {0, 1} random variables with mean ½) such that λ(Bi, Bj) = λij. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.

Multivariate Distributions with Fixed Marginals and Correlations

Consider the problem of drawing random variates (X 1, …, X n ) from a distribution where the marginal of each X i is specified, as well as the correlation between every pair X i and X j . For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between X i and X j . Any achievable correlation between X i and X j is a convex combination of these bounds. We call the value λ(X i , X j ) ∈ [0, 1] of this convex combination the convexity parameter of (X i , X j ) with λ(X i , X j ) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F 1, …, F n of (X 1, …, X n ), we show that λ(X i , X j ) = λ ij if and only if there exist symmetric Bernoulli random variables (B 1, …, B n ) (that is {0, 1} random variables with mean ½) such that λ(B i , B j ) = λ ij . In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.

Effects of leaf development and phosphorus supply on the photosynthetic characteristics of perennial legume species with pasture potential: modelling photosynthesis with leaf development

Age-dependent changes in leaf photosynthetic characteristics (i.e. parameters of the light response curve (maximum photosynthetic rate (Pmax), quantum yield (Φ) and the convexity parameter (θ)), stomatal conductance (gs) and dark respiration rate (Rd)) of an exotic perennial legume, Medicago sativa L. (lucerne), and two potential pasture legumes native to Australia, Cullen australasicum (Schltdl.) J.W. Grime and Cullen pallidum A. Lee, grown in a glasshouse for 5 months at two phosphorus (P) levels (3 (P3) and 30 (P30) mg P kg–1 dry soil) were tested. Leaf appearance rate and leaf area were lower at P3 than at P30 in all species, with M. sativa being the most sensitive to P3. At any leaf age, photosynthetic characteristics did not differ between P treatments. However, Pmax and gs for all the species and Φ for Cullen species increased until full leaf expansion and then decreased. The convexity parameter, θ, did not change with leaf age, whereas Rd decreased. The estimates of leaf net photosynthetic rate (Pleaf) obtained through simulations at variable Pmax and Φ were lower during early and late leaf developmental stages and at lower light intensities than those obtained when Φ was assumed to be constant (e.g. for a horizontally placed leaf, during the 1500°C days developmental period, 3 and 19% reduction of Pleaf at light intensities of 1500 and 500 µmol m–2 s–1, respectively). Therefore, developmental changes in leaf photosynthetic characteristics should be considered when estimating and simulating Pleaf of these pasture species.

THE ISGUR–WISE FUNCTION IN A QCD INSPIRED QUARK MODEL

We summarize the results of the Isgur–Wise function investigated within a QCD inspired quark model. We estimate the slope and the convexity parameter of the function.