Convex relaxation for solving posynomial programs

Convex underestimation techniques for nonlinear functions are an essential part of global optimization. These techniques usually involve the addition of new variables and constraints. In the case of posynomial functions \({x_1^{\alpha _1 } x_2^{\alpha _2 }\ldots x_n^{\alpha _n } ,}\) logarithmic transformations (Maranas and Floudas, Comput. Chem. Eng. 21:351–370, 1997) are typically used. This study develops an effective method for finding a tight relaxation of a posynomial function by introducing variables y _j and positive parameters β _j , for all α _j > 0, such that \({y_j =x_j^{-\beta _j }}\) . By specifying β _j carefully, we can find a tighter underestimation than the current methods.     

Tight convex underestimators for $${\mathcal{C}^2}$$ -continuous problems: II. multivariate functions

In Part I (Gounaris, C.E., Floudas, C.A.: Tight convex understimators for -continuous functions: I: Univariate functions. J. Global Optim. (2008). doi: ), we introduced a novel approach for the underestimation of univariate functions which was based on a piecewise application of the well-known αBB underestimator. The resulting underestimators were shown to be very tight and, in fact, can be driven to coincide with the convex envelopes themselves. An approximation by valid linear supports, resulting in piecewise linear underestimators was also presented. In this paper, we demonstrate how one can make use of the high quality results of the approach in the univariate case so as to extend its applicability to functions with a higher number of variables. This is achieved by proper projections of the multivariate αBB underestimators into select two-dimensional planes. Furthermore, since our method utilizes projections into lower-dimensional spaces, we explore ways to recover some of the information lost in this process. In particular, we apply our method after having transformed the original problem in an orthonormal fashion. This leads to the construction of even tighter underestimators, through the accumulation of additional valid linear cuts in the relaxation.     

Generation of networks with prescribed degree-dependent clustering

We propose a systematic, rigorous mathematical optimization methodology for the construction, “on demand,” of network structures that are guaranteed to possess a prescribed collective property: the degree-dependent clustering. The ability to generate such realizations of networks is important not only for creating artificial networks that can perform desired functions, but also to facilitate the study of networks as part of other algorithms. This problem exhibits large combinatorial complexity and is difficult to solve with off-the-shelf commercial optimization software. To that end, we also present a customized preprocessing algorithm that allows us to judiciously fix certain problem variables and, thus, significantly reduce computational times. Results from the application of the framework to data sets resulting from simulations of an acquaintance network formation model are presented.     

A preface to the special issue in memory of Professor Christodoulos A. Floudas

Optimization Letters -     

Stereochemically consistent reaction mapping and identification of multiple reaction mechanisms through integer linear optimization

Reaction mappings are of fundamental importance to researchers studying the mechanisms of chemical reactions and analyzing biochemical pathways. We have developed an automated method based on integer linear optimization, ILP, to identify optimal reaction mappings that minimize the number of bond changes. An alternate objective function is also proposed that minimizes the number of bond order changes. In contrast to previous approaches, our method produces mappings that respect stereochemistry. We also show how to locate multiple reaction mappings efficiently and determine which of those mappings correspond to distinct reaction mechanisms by automatically detecting molecular symmetries. We demonstrate our techniques through a number of computational studies on the GRI-Mech, KEGG LIGAND, and BioPath databases. The computational studies indicate that 99% of the 8078 reactions tested can be addressed within 1 CPU hour. The proposed framework has been incorporated into the Web tool DREAM ( http://selene.princeton.edu/dream/ ), which is freely available to the scientific community.     

Computational characterization of zeolite porous networks: an automated approach

An automated method has been developed to fully characterize the three-dimensional structure of zeolite porous networks. The proposed optimization-based approach starts with the crystallographic coordinates of a structure and identifies all portals, channels, and cages in a unit cell, as well as their connectivity. We apply our algorithms to known zeolites, hypothetical zeolites, and zeolite-like structures and use the characterizations to calculate important quantities such as pore size distribution, accessible volume, surface area, and largest cavity and pore limiting diameters. We aggregate this data over many framework types to gain insights about zeolite selectivity. Finally, we develop a continuous-time Markov chain model to estimate the probability of occupancy of adsorption sites throughout the porous network. ZEOMICS, an online database of structure characterizations and web tool for the automated approach is freely available to the scientific community (http://helios.princeton.edu/zeomics/).     

Tight convex underestimators for {{\mathcal C}^2}-continuous problems: I. univariate functions

A novel method for the convex underestimation of univariate functions is presented in this paper. The method is based on a piecewise application of the well-known αBB underestimator, which produces an overall underestimator that is piecewise convex. Subsequently, two algorithms are used to identify the linear segments needed for the construction of its -continuous convex envelope, which is itself a valid convex underestimator of the original function. The resulting convex underestimators are very tight, and their tightness benefits from finer partitioning of the initial domain. It is theoretically proven that there is always some finite level of partitioning for which the method yields the convex envelope of the function of interest. The method was applied on a set of univariate test functions previously presented in the literature, and the results indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.     

Generalized Hose uncertainty in single-commodity robust network design

Single-commodity network design considers an edge-weighted, undirected graph with a supply/demand value at each node. It asks for minimum weight capacities     

Systematic Analysis of Polymer Molecular Weight Influence on the Organic Photovoltaic Performance

The molecular weight of an electron donor‐conjugated polymer is as essential as other well‐known parameters in the chemical structure of the polymer, such as length and the nature of any side groups (alkyl chains) positioned on the polymeric backbone, as well as their placement, relative strength, the ratio of the donor and acceptor moieties in the backbone of donor–acceptor (D–A)‐conjugated polymers, and the arrangement of their energy levels for organic photovoltaic performance. Finding the “optimal” molecular weight for a specific conjugated polymer is an important aspect for the development of novel photovoltaic polymers. Therefore, it is evident that the chemistry of functional conjugated polymers faces major challenges and materials have to adopt a broad range of specifications in order to be established for high photovoltaic performance. In this review, the approaches followed for enhancing the molecular weight of electron‐donor polymers are presented in detail, as well as how this influences the optoelectronic properties, charge transport properties, structural conformation, morphology, and the photovoltaic performance of the active layer.