Mixture Selection, Mechanism Design, and Signaling

We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function $g$ from the $n$-dimensional hypercube to the bounded interval $[-1,1]$, and an $n \times m$ matrix $A$ with bounded entries, maximize $g(Ax)$ over $x$ in the $m$-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. signaling). We present an approximation algorithm for this problem when $g$ simultaneously satisfies two smoothness properties: Lipschitz continuity with respect to the $L^\infty$ norm, and noise stability. The latter notion, which we define and cater to our setting, controls the degree to which low-probability errors in the inputs of $g$ can impact its output. When $g$ is both $O(1)$-Lipschitz continuous and $O(1)$-stable, we obtain an (additive) PTAS for mixture selection. We also show that neither assumption suffices by itself for an additive PTAS, and both assumptions together do not suffice for an additive FPTAS. We apply our algorithm to different game-theoretic applications from mechanism design and optimal signaling. We make progress on a number of open problems suggested in prior work by easily reducing them to mixture selection: we resolve an important special case of the small-menu lottery design problem posed by Dughmi, Han, and Nisan; we resolve the problem of revenue-maximizing signaling in Bayesian second-price auctions posed by Emek et al. and Miltersen and Sheffet; we design a quasipolynomial-time approximation scheme for the optimal signaling problem in normal form games suggested by Dughmi; and we design an approximation algorithm for the optimal signaling problem in the voting model of Alonso and C\^{a}mara

Designing optimal mixtures using generalized disjunctive programming: Hull relaxations

A general modeling framework for mixture design problems, which integrates Generalized Disjunctive Programming (GDP) into the Computer-Aided Mixture/blend Design (CAMbD) framework, was recently proposed (S. Jonuzaj, P.T. Akula, P.-M. Kleniati, C.S. Adjiman, 2016. AIChE Journal 62, 1616–1633). In this paper we derive Hull Relaxations (HR) of GDP mixture design problems as an alternative to the big-M (BM) approach presented in this earlier work. We show that in restricted mixture design problems, where the number of components is fixed and their identities and compositions are optimized, BM and HR formulations are identical. For general mixture design problems, where the optimal number of mixture components is also determined, a generic approach is employed to enable the derivation and solution of the HR formulation for problems involving functions that are not defined at zero (e.g., logarithms). The design methodology is applied successfully to two solvent design case studies: the maximization of the solubility of a drug and the separation of acetic acid from water in a liquid-liquid extraction process. Promising solvent mixtures are identified in both case studies. The HR and BM approaches are found to be effective for the formulation and solution of mixture design problems, especially via the general design problem

The D-optimal design of blocked and split-plot experiments with mixture components.

So far, the optimal design of blocked and split-plot experiments involving mixture components has received scant attention. In this paper, an easy method to construct efficient blocked mixture experiments in the presence of fixed and/or random blocks is presented. The method can be used when qualitative variables are involved in a mixture experiment as well. It is also shown that orthogonally blocked mixture experiments are highly inefficient compared to D-optimal designs. Finally, the design of a split-plot mixture experiment with process variables is discussed.Design; Fixed and random blocks; Minimum support design; Mixture experiment; Optimal; Optimal design; Orthogonal blocking; Process variables; Processes; Qualitative variables; Split-plot experiment; Variables;

Extensions of D-optimal Minimal Designs for Symmetric Mixture Models.

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide desirable response characteristics in finished products. D-optimal minimal designs have been considered for a variety of mixture models, including Scheffé\u27s linear, quadratic, and cubic models. Usually, these D-optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the Lack of Fit tests. Also, the majority of the design points in D-optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. IN THIS PAPER EXTENSIONS OF THE D-OPTIMAL MINIMAL DESIGNS ARE DEVELOPED FOR A GENERAL MIXTURE MODEL TO ALLOW ADDITIONAL INTERIOR POINTS IN THE DESIGN SPACE TO ENABLE PREDICTION OF THE ENTIRE RESPONSE SURFACE: Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. We compare the proposed designs with Cornell (1986) two ten-point designs for the Lack of Fit test by simulations

Additive mixture effects of estrogenic chemicals in human cell-based assays can be influenced by inclusion of chemicals with differing effect profiles

Copyright @ 2012 The Authors. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and 85 reproduction in any medium, provided the original author and source are credited. The article was made available through the Brunel University Open Access Publishing Fund.A growing body of experimental evidence indicates that the in vitro effects of mixtures of estrogenic chemicals can be well predicted from the estrogenicity of their components by the concentration addition (CA) concept. However, some studies have observed small deviations from CA. Factors affecting the presence or observation of deviations could include: the type of chemical tested; number of mixture components; mixture design; and assay choice. We designed mixture experiments that address these factors, using mixtures with high numbers of components, chemicals from diverse chemical groups, assays with different in vitro endpoints and different mixture designs and ratios. Firstly, the effects of mixtures composed of up to 17 estrogenic chemicals were examined using estrogenicity assays with reporter-gene (ERLUX) and cell proliferation (ESCREEN) endpoints. Two mixture designs were used: 1) a 'balanced' design with components present in proportion to a common effect concentration (e.g. an EC(10)) and 2) a 'non-balanced' design with components in proportion to potential human tissue concentrations. Secondly, the individual and simultaneous ability of 16 potential modulator chemicals (each with minimal estrogenicity) to influence the assay outcome produced by a reference mixture of estrogenic chemicals was examined. Test chemicals included plasticizers, phthalates, metals, PCBs, phytoestrogens, PAHs, heterocyclic amines, antioxidants, UV filters, musks, PBDEs and parabens. In all the scenarios tested, the CA concept provided a good prediction of mixture effects. Modulation studies revealed that chemicals possessing minimal estrogenicity themselves could reduce (negatively modulate) the effect of a mixture of estrogenic chemicals. Whether the type of modulation we observed occurs in practice most likely depends on the chemical concentrations involved, and better information is required on likely human tissue concentrations of estrogens and of potential modulators. Successful prediction of the effects of diverse chemical combinations might be more likely if chemical profiling included consideration of effect modulation.This study is funded by the United Kingdom Food Standards Agency (Contract Number T01045)

Solid-Liquid Equilibria Modelling for triacylglycerols exhibiting multiple solid phases

Important end-use properties of vegetable oils based edible products are strong related to the equilibrium between a solid crystalline network and a liquid phase. This work presents the available literature development of solid-liquid equilibrium in triacylglycerol systems and highlights how it can be coupled with a Computer-Aided Mixture and Blend Design framework, for design new mixtures/blends with improved properties allowing a better use of renewable resources as vegetable oils. Stability tests were implemented as they are an essential step for powerful solid liquid equilibrium resolution and some results were presented for a four component triacylglycerol mixture in different temperatures and composition

Propulsion sizing program

Computer program was written to evaluate and define optimum design parameters of low pressure auxiliary propulsion system. APS will provide attitude and translational control of space shuttle vehicles. Program evaluates subsystem design parameters such as engine chamber pressure, mixture ratio, expansion ratio, and subsystem component weight and size