Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson’s outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson’s algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems

Non-Convex Optimization and Applications to Bilinear Programming and Super-Resolution Imaging

Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering applications, economics, and operations research, and have a natural encoding to quadratic programs. They appear in the study of Lyapunov functions used to deduce the stability of solutions to differential equations describing dynamical systems. For multivariate dynamical systems, the problem formulation for computing an appropriate Lyapunov function is a BLP. In electric power systems engineering, one of the most practically important and well-researched subfields of constrained nonlinear optimization is Optimal Power Flow wherein one attempts to optimize an electric power system subject to physical constraints imposed by electrical laws and engineering limits, which can be naturally formulated as a quadratic program. In a recent publication, we studied the relationship between data flow constraints for numerical domains such as polyhedra and bilinear constraints. The problem of recovering an image from its Fourier modulus, or intensity, measurements emerges in many physical and engineering applications. The problem is known as Fourier phase retrieval wherein one attempts to recover the phase information of a signal in order to accurately reconstruct it from estimated intensity measurements by applying the inverse Fourier transform. The problem of recovering phase information from a set of measurements can be formulated as a quadratic program. This problem is well-studied but still presents many challenges. The resolution of an optical device is defined as the smallest distance between two objects such that the two objects can still be recognized as separate entities. Due to the physics of diffraction, and the way that light bends around an obstacle, the resolving power of an optical system is limited. This limit, known as the diffraction limit, was first introduced by Ernst Abbe in 1873. Obtaining the complete phase information would enable one to perfectly reconstruct an image; however, the problem is severely ill-posed and the leads to a specialized type of quadratic program, known as super-resolution imaging, wherein one attempts to learn phase information beyond the limits of diffraction and the limitations imposed by the imaging device

Non-Convex Optimization and Applications to Bilinear Programming and Super-Resolution Imaging

Bilinear programs and Phase Retrieval are two instances of nonconvex problems that arise in engineering and physical applications, and both occur with their fundamental difficulties. In this thesis, we consider various methods and algorithms for tackling these challenging problems and discuss their effectiveness. Bilinear programs (BLPs) are ubiquitous in engineering applications, economics, and operations research, and have a natural encoding to quadratic programs. They appear in the study of Lyapunov functions used to deduce the stability of solutions to differential equations describing dynamical systems. For multivariate dynamical systems, the problem formulation for computing an appropriate Lyapunov function is a BLP. In electric power systems engineering, one of the most practically important and well-researched subfields of constrained nonlinear optimization is Optimal Power Flow wherein one attempts to optimize an electric power system subject to physical constraints imposed by electrical laws and engineering limits, which can be naturally formulated as a quadratic program. We study the relationship between data flow constraints for numerical domains such as polyhedra and bilinear constraints. The problem of recovering an image from its Fourier modulus, or intensity, measurements emerges in many physical and engineering applications. The problem is known as Fourier phase retrieval wherein one attempts to recover the phase information of a signal in order to accurately reconstruct it from estimated intensity measurements by applying the inverse Fourier transform. The problem of recovering phase information from a set of measurements can be formulated as a quadratic program. This problem is well-studied but still presents many challenges. The resolution of an optical device is defined as the smallest distance between two objects such that the two objects can still be recognized as separate entities. Due to the physics of diffraction, and the way that light bends around an obstacle, the resolving power of an optical system is limited. This limit, known as the diffraction limit, was first introduced by Ernst Abbe in 1873. Obtaining the complete phase information would enable one to perfectly reconstruct an image; however, the problem is severely ill-posed and the leads to a specialized type of quadratic program, known as super-resolution imaging, wherein one attempts to learn phase information beyond the limits of diffraction and the limitations imposed by the imaging device.</p

Unemployment, institutions and reform complementarities: Re-assessing the aggregate evidence for OECD countries

There is no or limited consensus on the quantitative impact of institutions on unemployment, which has led some to question the case for structural reforms. Recent studies suggest also that institutions interact with each other and cannot be analysed in isolation. In this paper, we estimate a standard reduced-form model to explore the institutional determinants of unemployment and assess its robustness using a large battery of robustness checks. We show that, although the impact of each individual policy varies across countries due to policy interactions, the simple linear model can be used to draw inferences for countries with an average mix of institutions. The model is then extended to encompass systemic interactions, in which individual policies interact with the overall institutional framework. We find relatively robust evidence of broad reform complementarities.institutions; aggregate unemployment; reform complementarities

Characterizations of long-run producer optima and the short-runapproach to long-run market equilibrium: a general theory withapplications to peak-load pricing

This is a new formal framework for the theory of competitive equilibrium and its applications.Our "short-run approach" means the calculation of long-run producer optimaand general equilibria from the short-run solutions to the producer's profit maximizationprogramme and its dual. The marginal interpretation of the dual solution means that itcan be used to value the capital and other fixed inputs, whose levels are then adjustedaccordingly (where possible). But short-run profit can be a nondifferentiable function ofthe fixed quantities, and the short-run cost is nondifferentiable whenever there is a rigidcapacity constraint. Nondifferentiability of the optimal value requires the introductionof nonsmooth calculus into equilibrium analysis, and subdifferential generalizations ofsmooth-calculus results of microeconomics are given, including the key Wong-Viner EnvelopeTheorem. This resolves long-standing discrepancies between "textbook theory"and industrial experience. The other tool employed to characterise long-run produceroptima is a primal-dual pair of programmes. Both marginalist and programming characterizationsof producer optima are given in a taxonomy of seventeen equivalent systemsof conditions. When the technology is described by production sets, the most usefulsystem for the short-run approach is that using the short-run profit programme andits dual. This programme pair is employed to set up a formal framework for long-rungeneral-equilibrium pricing of a range of commodities with joint costs of production.This gives a practical method that finds the short-run general equilibrium en route tothe long-run equilibrium, exploiting the operating policies and plant valuations that mustbe determined anyway. These critical short-run solutions have relatively simple formsthat can greatly ease the fixed-point problem of solving for equilibrium, as is shownon an electricity pricing example. Applicable criteria are given for the existence of theshort-run solutions and for the absence of a duality gap. The general analysis is speltout for technologies with conditionally fixed coefficients, a concept extending that of thefixed-coefficients production function to the case of multiple outputs. The short-run approachis applied to the peak-load pricing of electricity generated by thermal, hydro andpumped-storage plants. This gives, for the first time, a sound method of valuing thefixed assets-in this case, river flows and the sites suitable for reservoirs.general equilibrium, fixed-input valuation, nondifferentiable joint costs,Wong-Viner Envelope Theorem, public utility pricing

Measurements, Models, Systems and Design

531 s.