Bounds tightening based on optimality conditions for nonconvex box-constrained optimization

First-order optimality conditions have been extensively studied for the development of algorithms for identifying locally optimal solutions. In this work, we propose two novel methods that directly exploit these conditions to expedite the solution of box-constrained global optimization problems. These methods carry out domain reduction by application of bounds tightening methods on optimality conditions. This scheme is implicit and avoids explicit generation of optimality conditions through symbolic differentation, which can be memory and time intensive. The proposed bounds tightening methods are implemented in the global solver BARON. Computational results on a test library of 327 problems demonstrate the value of our proposed approach in reducing the computational time and number of nodes required to solve these problems to global optimality.     

Radon—Nikodym theorems for multimeasures and transition multimeasures

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Research supported by N. S. F. Grant DMS-8802688.     

Bifurcation near infinity for the Neumann problem with concave–convex nonlinearities

In this Note, we study a class of Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing terms (concave–convex nonlinearities). Using the Ambrosetti–Rabinowitz condition and related topological and variational arguments, we prove a bifurcation result for large values of the parameter.     

Global optimization of general nonconvex problems with intermediate polynomial substructures

This work considers the global optimization of general nonconvex nonlinear and mixed-integer nonlinear programming problems with underlying polynomial substructures. We incorporate linear cutting planes inspired by reformulation-linearization techniques to produce tight subproblem formulations that exploit these underlying structures. These cutting plane strategies simultaneously convexify linear and nonlinear terms from multiple constraints and are highly effective at tightening standard linear programming relaxations generated by sequential factorable programming techniques. Because the number of available cutting planes increases exponentially with the number of variables, we implement cut filtering and selection strategies to prevent an exponential increase in relaxation size. We introduce algorithms for polynomial substructure detection, cutting plane identification, cut filtering, and cut selection and embed the proposed implementation in BARON at every node in the branch-and-bound tree. A computational study including randomly generated problems of varying size and complexity demonstrates that the exploitation of underlying polynomial substructures significantly reduces computational time, branch-and-bound tree size, and required memory.     

Multiple nontrivial solutions for nonlinear periodic problems with the p-Laplacian

We consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential (hemivariational inequality). Using the degree theory for multivalued perturbations of (S)₊-operators and the spectrum of a class of weighted eigenvalue problems for the scalar p-Laplacian, we prove the existence of at least three distinct nontrivial solutions, two of which have constant sign.     

The string coupling accelerates the expansion of the universed

Generic cosmological models in non-critical string theory have a time-dependent dilaton background at a late epoch. The cosmological deceleration parameter q ₀ is given by the square of the string coupling, g _s ², up to a negative sign. Hence the expansion of the Universe must accelerate eventually, and the observed value of q ₀ corresponds to g _s ² ∼ 0.6. In this scenario, the string coupling is asymptotically free at large times, but its present rate of change is imperceptibly small. First Award in the 2005 Essay Competition of the Gravity Research Foundation. - Ed.     

Tagging of the Magnetization with the Transition Zones of 360° Rotations Generated by a Tandem of Two Adiabatic DANTE Inversion Sequences

A new technique is presented for generating myocardial tagging using the signal intensity minima of the transition zones between the bands of 0° and 360° rotations, induced by a tandem of two adiabatic delays alternating with nutations for tailored excitation (DANTE) inversion sequences. With this approach, the underlying matrix corresponds to magnetization that has experienced 0° or 360° rotations. The DANTE sequences were implemented from adiabatic parent pulses for insensitivity of the underlying matrix to B₁ inhomogeneity. The performance of the proposed tagging technique is demonstrated theoretically with computer simulations and experimentally on phantom and on the canine heart, using a surface coil for both RF transmission and signal reception. The simulations and the experimental data demonstrated uniform grid contrast and sharp tagging profiles over a twofold variation of the B₁ field magnitude.     

The swelling interface number as a criterion for prediction of diffusional solute release mechanisms in swellable polymers

When a glassy polymer containing a uniformly dispersed solute is brought in contact with a penetrant, solute diffusion will be associated with the transport mechanism and penetration velocity of the penetrant in the polymer. Analysis and prediction of mechanisms of diffusional solute release may be obtained through a new dimensionless number, the swelling interface number, Sw, which compares the relative mobilities of the penetrant and the solute in the presence of macromolecular relaxations in the polymer. It is shown that a sufficient and necessary criterion for time-independent diffusional solute release rates from these swellable systems is that the Sw be smaller than 10^?2. The swelling interface number Sw may be related to easily determined structural and thermodynamic parameters of the solute/polymer/penetrant system. Preliminary experimental results of dynamic water swelling of poly(2-hydroxyethyl methacrylate-co-methyl methacrylate) and diffusional release of theophylline from initially glassy copolymers show that decreasing values of Sw are related to increased pseudo-case-II transport kinetics of the solute.     

Modelling of drug diffusion through swellable polymeric systems

A drug diffusion model for the case of diffusion of an initially uniformly distributed drug through a polymeric matrix is presented and solved. Drug diffusion from a single surface is analyzed for the case of countercurrent diffusion of a solvent which is thermodynamically compatible with the polymer. Due to swelling, considerable volume expansion is observed leading to a moving-boundary diffusion problem. Drug concentration profiles within the polymer and drug release rates can be determined. The results are in agreement with experimental data obtained for the system of KCl distributed in hydrophilic hydroxypropyl methyl cellulose matrices, in the form of tablets.