A method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

First-order strong variation algorithms for optimal control problems with terminal inequality constraints

Journal of Optimization Theory and Applications - This paper presents eight algorithms for solving optimal control problems with general constraints on the control and inequality constraints on the...     

Tracer transport in a fractured chalk: X-ray CT characterization and digital-image-based (DIB) simulation

A digital-image-based simulation methodology is applied to evaluate the influence of heterogeneous porosity on the evolution of tracer concentrations in imaged tracer tests. Maps of computed tomography (CT)-number are calibrated relative to average porosity, and then thresholded to define porosity maps. These data are then used to automate the distribution of parameters within a finite element representation of the geometry. The technique is applied to characterize the variability of the porosity, the hydraulic conductivity, and the diffusivity for an artificially fractured chalk core (30 × 5 cm). X-ray CT was used both to characterize the initial condition of the core, and then to concurrently monitor the transport of an NaI tracer within the fracture and into the surrounding matrix. The X-ray CT imaging is used to characterize the heterogeneous rock porosity, based on which the hydraulic conductivity, and diffusivity of the chalk were defined and were directly imported into our newly developed three-dimensional FEMLAB-based multiple physics simulator. Numerical simulations have confirmed the observed tracer transport behaviors: (1) The different tracer-penetration distances imaged in the matrix above and below the horizontal fracture are indicative of a greater tracer mass penetrating into the lower matrix; and (2) Transport in the matrix below the fracture was enhanced. The computer simulated tracer concentration distributions compare favorably with those monitored by X-ray CT.     

Die Verstärkung der Prädissoziation durch Stöße und das Beersche Gesetz

Zeitschrift für Physik A Hadrons and nuclei - Es wurde durch die Aufnahme des Absorptionsspektrums von gasförmigem Brom bei verschiedenen Drucken und Schichtdicken festgestellt, daß...     

Superlinearly convergent algorithm for min-max problems

Algorithms for solving the problem of minimizing the maximum of a finite number of functions are proposed and analyzed. Quadratic approximations to the functions are employed in the determination of a search direction. Global convergence is proven and it is shown that a quadratic rate of convergence is obtained.This research was sponsored in part by National Science Foundation Grant ECS-81-21149, Air Force Office of Scientific Research Grant 83-0361, Office of Naval Research Contract N00014-83-K-0602, Semiconductor Research Corporation Contract SRC-82-11-008, State of California MICRO Program, and General Electric Company.     

Size dependence of freezing temperature and structure instability in simulated Lennard-Jones clusters

Liquid Lennard-Jones clusters of 14 different sizes from N=55 to 923 particles were cooled down to find their temperature of liquid-solid transition and the internal structure of the solidified clusters. The decrease of the cluster temperature was attained by a gradual change of the system temperature in Monte Carlo simulations. The liquid-to-solid transition was found by analysis of the specific heat as well as by detection of the structural units of face-centred cubic, hexagonal close-packed and decahedral type. It was observed that near the detected transition temperature the solid-like cluster structure is not always stable and fluctuates between solid and liquid states. The fluctuations of the state were observed frequently for small clusters with N ≤147, where the temporary solid structure is created by a large part of internal atoms. Manual inspection of cluster structural data and the 10%N condition for minimal number of atoms as centres of solid-like units enable detection of stable cluster solidification at freezing temperature. It was found that the freezing temperature of all clusters, with the exception of N=55, decreases linearly with N^-1/3. The extrapolated freezing temperature of the bulk LJ system is 13% lower than the experimental value of argon. After freezing, the solid phase remains but some atoms close to the cluster surface are not firmly included into the structure and oscillate mainly between solid structure and disordered one.     

A generalized quadratic programming-based phase I-phase II method for inequality-constrained optimization

We present a globally convergent phase I-phase II algorithm for inequality-constrained minimization, which computes search directions by approximating the solution to a generalized quadratic program. In phase II these search directions are feasible descent directions. The algorithm is shown to converge linearly under convexity assumptions. Both theory and numerical experiments suggest that it generally converges faster than the Polak-Trahan-Mayne method of centers.The research reported herein was sponsored in part by the Air Force Office of Scientific Research (Grant AFOSR-90-0068), the National Science Foundation (Grant ECS-8713334), and a Howard Hughes Doctoral Fellowship (Hughes Aircraft Co.).