An ODE-based approach to nonlinearly constrained minimax problems

We consider the following problem: Choosex ₁, ...,x _n to wherem ₁,m ₂,m ₃ are integers with 0m ₁m ₂m ₃, thef _i are given real numbers, and theg _i are given smooth functions. Constraints of the formg _i (x ₁, ...,x _n )=0 can also be handled without problem. Each iteration of our algorithm involves approximately solving a certain non-linear system of first-order ordinary differential equations to get a search direction for a line search and using a Newton-like approach to correct back into the feasible region when necessary. The algorithm and our Fortran implementation of it will be discussed along with some examples. Our experience to date has been that the program is more robust than any of the library routines we have tried, although it generally requires more computer time. We have found this program to be an extremely useful tool in diverse areas, including polymer rheology, computer vision, and computation of convexity-preserving rational splines.     

Determination of the intrinsic site pK(a) value and cooperativity of the symmetrical hexadentate chelator N,N',N'-tris[2-(3-hydroxy-2-oxo-1,2-dihydropyridin-1-yl)acetamido]ethylamine

The three overlapping pK(a) values of N,N',N'-tris[2-(3-hydroxy-2-oxo-1,2-dihydropyridin-1-yl)acetamido]ethylamine, a tripodal hexadentate chelator formed from three 3-hydroxy-2(1H)-pyridinone moieties amide linked to tris-(2-aminoethyl)amine, were determined by simultaneous spectrophotometric and potentiometric titration. The data was analysed by non-linear regression with constraints to deal with (a) the highly correlated absorptivities and (b) the highly correlated pK(a) values. The three pK(a) values were optimized first from the spectrophotometric data (absorbance vs. pH) by non-linear regression to a model in which the molar absorptivity of the ith species ((i)) was constrained by the correlation equation (i) = epsilon (0) + (epsilon (3) - epsilon (0))i 3 with i = 0, 1, 2, 3, where (3) and (0) represent the molar absorptivities of the most protonated and least protonated species, respectively. The molar absorbitivity of the four species defined by three pK(a) values is, therefore, linearly related to proton stoichiometry. The pK(a) values were then optimized from the potentiometric data (pH vs. titrant volume) by non-linear regression to a model in which the three pK(a) values were constrained by the correlation equation pK(a(i)) = pK(a(int)) + b(i - 1) + (i - 2)log(3) where i = 1, 2 or 3. This expresses the three pK(a) values in terms of only two optimizable parameters, the intrinsic site pK(a) (pK(a(int))) and the interaction energy between sites (b). The fixed term (i - 2)log(3) accounts for the statistical effect on the pK(a) values of three equivalent ionizable sites. The modified analytical derivatives required for optimization of these parameters by the Gauss-Newton-Marquardt algorithm and the merits of optimizing pK(a) values with these two correlation equations are discussed. The optimized pK(a) values were 9.31 +/- 0.01, 8.75 +/- 0.01 and 8.19 +/- 0.01. The separation between pK(a) values is 0.58 comprising 0.477 for the statistical effect and 0.081 for the interaction energy while the intrinsic site pK(a) is 8.672 +/- 0.005. The tertiary amine at the centre of the tripodal backbone has a pK(a) of 5.88 +/- 0.03.