Generalized Stationary Points and an Interior Point Method for MPEC

Mathematical program with equilibrium constraints (MPEC)has extensive applications in practical areas such as traffic control, engineering design, and economic modeling. Some generalized stationary points of MPEC are studied to better describe the limiting points produced by interior point methods for MPEC.A primal-dual interior point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced without assuming strict complementarity or linear independence constraint qualification. Under very general assumptions, the algorithm can always find some point with strong or weak stationarity. In particular, it is shown that every limiting point of the generated sequence is a piece-wise stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a certain point with weak stationarity can be obtained. Preliminary numerical results are satisfactory, which include a case analyzed by Leyffer for which the penalty interior point algorithm failed to find a stationary solution.Singapore-MIT Alliance (SMA

Nemlineáris egyensúlyi rendszerek elméleti és módszertani kérdései = Theoretical and methodological issues of nonlinear equilibrium systems

A nemlineáris egyensúlyi rendszerek területén új eredményeket értünk el egyes feladatosztályok megoldhatóságának skaláris deriváltakra alapozott jellemzésével. A nemlineáris egyensúlyi rendszerek egy új megközelítését adtuk az izotón projekciós kúpok és kiterjesztéseik segítségével, és új eredményeket mutattunk be izoton projekciós kúpokkal történő rekurziók konvergenciáját illetően. Megmutattuk a metszetgörbülek nemlineáris optimalizálásban betöltött szerepét, és kvadratikus törtfüggvények speciális tulajdonságaival is foglalkoztunk. Fontos módszertani és implementációs eredményeket értünk el a kvadratikus optimalizálás belső pontos módszereinek területén is. Új blokkolási sémát fejlesztettünk ki a belső pontos algoritmusoknál előforduló szimmetrikus mátrixok faktorizációjához. A kvadratikus feltételek melletti konvex optimalizálás fontos feladatosztály a folytonos optimalizálásban. Megmutattuk, hogy belső pontos módszerekkel ez a feladatosztály nagy méretekben is hatékonyan kezelhető. Sikeresen alkalmaztuk a nemlineáris programozást döntési feladatok megoldásánál, elsősorban páros összhasonlítási mátrixok konzisztens márixokkal való közelítésével kapcsolatban. Új módszereket mutattunk be a legkisebb négyzetek célfüggvényű közelítés globális optimális megoldásainak meghatározására, valamint kiterjesztettük a sajátvektor módszert a nem teljesen kitöltött páros összehasonlítási mátrixok esetére | New results have been achieved in the field of nonlinear equilibrium problems by characterizing the solvability of some problem classes based on scalar derivatives. A new approach has been presented for the nonlinear equilibrium systems by the help of isotone projection cones and their extensions. Also, new results were presented on the convergence of recursions with isotone projection cones. We pointed out the role of sectional curvatures in nonlinear optimization. Some special properties of quadratic fractional functions have been also dealt with. We achieved important methodological and implementational results in the field of interior point methods of quadratic optimization. A new blocking scheme was developed for the symmetric matrix factorizations arising in interior point methods. An important class of the continuous optimization is that of the quadratically constrained convex problems. New techniques have been presented that improve the efficiency of interior point methods when solving quadratically constrained large-scale problems. Nonlinear programming was applied successfully at solving some decision problems, mainly at approximating pairwise comparison matrices by consistent ones. We presented new methods for finding the global optimal solutions in the case of approximating in the least squares sense. We also extended the eigenvector method for the case of incomplete pairwise comparison matrices

Computational methods for geochemical modelling: applications to carbon dioxide sequestration

Geochemical modelling is fundamental for solving many environmental problems, and specially useful for modelling carbon storage into deep saline aquifers. This is because the injected greenhouse gas perturbs the reservoir, causing the subsurface fluid to become acidic, and consequently increasing its reactivity with the formation rock. Assessment of the long term fate of carbon dioxide, therefore, requires accurate calculations of the geochemical processes that occur underground. For this, it is important to take into account the major water-gas-rock effects that play important roles during the gas storage and migration. These reactive processes can in general be formulated in terms of chemical equilibrium or chemical kinetics models. This work proposes novel numerical methods for the solution of multiphase chemical equilibrium and kinetics problems. Instead of adapting or improving traditional algorithms in the geochemical modelling literature, this work adopts an approach of abstracting the underlying mathematics from the chemical problems, and investigating suitable, modern and efficient methods for them in the mathematical literature. This is the case, for example, of the adaptation of an interior-point minimisation algorithm for the calculation of chemical equilibrium, in which the Gibbs energy of the system is minimised. The methods were developed for integration into reactive transport simulators, requiring them to be accurate, robust and efficient. These features are demonstrated in the manuscript. All the methods developed were applied to problems relevant to carbon sequestration in saline aquifers. Their accuracy was assessed by comparing, for example, calculations of pH and CO2 solubility in brines against recent experimental data. Kinetic modelling of carbon dioxide injection into carbonate and sandstone saline aquifers was performed to demonstrate the importance of accounting for the water-gas-rock effects when simulating carbon dioxide sequestration. The results demonstrated that carbonate rocks, for example, increase the potential of the subsurface fluid to dissolve even more mobile CO2.Open Acces

An interior-point method for mpecs based on strictly feasible relaxations.

An interior-point method for solving mathematical programs with equilibrium constraints (MPECs) is proposed. At each iteration of the algorithm, a single primaldual step is computed from each subproblem of a sequence. Each subproblem is defined as a relaxation of the MPEC with a nonempty strictly feasible region. In contrast to previous approaches, the proposed relaxation scheme preserves the nonempty strict feasibility of each subproblem even in the limit. Local and superlinear convergence of the algorithm is proved even with a less restrictive strict complementarity condition than the standard one. Moreover, mechanisms for inducing global convergence in practice are proposed. Numerical results on the MacMPEC test problem set demonstrate the fast-local convergence properties of the algorithm

An interior point algorithm for computing equilibria in economies with incomplete asset markets

Computing equilibria in general equilibria models with incomplete asset (GEI) markets is technically difficult. The standard numerical methods for computing these equilibria are based on homotopy methods. Despite recent advances in computational economics, much more can be done to enlarge the catalogue of techniques for computing GEI equilibria. This paper presents an interior-point algorithm that exploits the special structure of GEI markets. We prove that the algorithm converges globally at a quadratic rate, rendering it particularly effective in solving large-scale GEI economies. To illustrate its performance, we solve relevant examples of GEI market

Constraint interface preconditioning for topology optimization problems

The discretization of constrained nonlinear optimization problems arising in the field of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological ill-conditioning, strong nonlinearity and size. In this work we propose a methodology which brings together existing fast algorithms, namely, interior-point for the optimization problem and a novel substructuring domain decomposition method for the ensuing large-scale linear systems. The main contribution is the choice of interface preconditioner which allows for the acceleration of the domain decomposition method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com

Inverse Design Based on Nonlinear Thermoelastic Material Models Applied to Injection Molding

This paper describes an inverse shape design method for thermoelastic bodies. With a known equilibrium shape as input, the focus of this paper is the determination of the corresponding initial shape of a body undergoing thermal expansion or contraction, as well as nonlinear elastic deformations. A distinguishing feature of the described method lies in its capability to approximately prescribe an initial heterogeneous temperature distribution as well as an initial stress field even though the initial shape is unknown. At the core of the method, there is a system of nonlinear partial differential equations. They are discretized and solved with the finite element method or isogeometric analysis. In order to better integrate the method with application-oriented simulations, an iterative procedure is described that allows fine-tuning of the results. The method was motivated by an inverse cavity design problem in injection molding applications. Its use in this field is specifically highlighted, but the general description is kept independent of the application to simplify its adaptation to a wider range of use cases.Comment: 22 pages, 13 figure