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Evaluation of the Optimal Utility of Some Investment Projects with Irreversible Environmental Effects

PURE AND APPLIED GEOPHYSICS 2015年第1期172卷 149-165页

作者： Arregui, Inigo Vazquez, Carlos Univ A Corua Fac Informat Dept Math La Coruna 15071 Spain

In this work, the authors propose efficient numerical methods to solve mathematical models for different optimal investment problems with irreversible environmental effects. A relevant point is that both the benefits of the environment and the alternative project are uncertain. The cases with instantaneous and progressive transformation of the environment are addressed. In the first case, an augmented lagrangian active set (ALAS) algorithm combined with finite element methods are proposed as a more efficient technique for the numerical solution to the obstacle problem associated with a degenerated elliptic PDE. In the second case, the mathematical model can be split into two subsequent steps: first we solve numerically a set of parameter dependent boundary value problems (the parameter being the level of progressive transformation), and secondly an evolutive nonstandard obstacle problem is discretized, thus leading to an obstacle problem at each time step. Also, an ALAS algorithm is proposed at each time step. Numerical solutions are validated through qualitative properties theoretically proven in the literature for different examples.

关键词： Investment under uncertainty obstacle problems finite elements augmented lagrangian algorithms

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IMAGE RESTORATION WITH MIXED OR UNKNOWN NOISES

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MULTISCALE MODELING & SIMULATION 2014年第2期12卷 458-487页

作者： Gong, Zheng Shen, Zuowei Toh, Kim-Chuan Natl Univ Singapore Dept Math Singapore 117576 Singapore Natl Univ Singapore Dept Math Singapore 119076 Singapore Singapore MIT Alliance Singapore 117576 Singapore

This paper proposes a simple model for image restoration with mixed or unknown noises. It can handle image restoration without assuming any prior knowledge of the noise distribution. It is particularly useful for solving real-life image restoration problems, since under various constraints, images are always degraded with mixed noise and it is impossible to determine what type of noise is involved. The proposed model can remove mixed-type noises as well as unknown noises and at the same time also works comparably well against the model whose data fitting term is designed for a specific given noise type. While most of the existing methods for image restoration are designed specifically for a given type of noise, ours appears to be the first universal model for handling image restoration with various mixed noises and unknown noises. Extensive simulations on synthetic data show that our method is effective and robust in restoring images contaminated by additive Gaussian noise, Poisson noise, random-valued impulse noise, multiplicative Gamma noise, and mixtures of these noises. Numerical results on real data show that it can remove noises without any prior knowledge of the noise distribution.

关键词： image restoration mixed-type noises L1-regularized convex minimization accelerated proximal gradient algorithms augmented lagrangian algorithms

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An augmented lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampere equation in two dimensions

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2006年 22卷 71-96页

作者： Dean, EJ Glowinski, R Univ Houston Dept Math Houston TX 77204 USA

In this article, we discuss the numerical solution of the Dirichlet problem for the real elliptic Monge-Ampere equation, in two dimensions, by an augmented lagrangian based iterative method. To derive the above algorithm, we take advantage of a reformulation of the Monge-Ampere problem as a saddle-point one, for a well-chosen augmented lagrangian functional over the product of appropriate primal and dual sets. The convergence of the finite element approximation and of the iterative methods described in this article still has to be proved, however, on the basis of numerical experiments reported in this article, it is safe to say that: ( i) The augmented lagrangian methodology discussed here provides a sequence converging to a solution of the Monge-Ampere problem under consideration, if such a solution exists in the space H-2(Omega). (ii) If, despite the smoothness of its data, the above problem has no solution, the above augmented lagrangian method solves it in a least-squares sense, to be precisely fined in this article.

关键词： elliptic Monge-Ampere equation augmented lagrangian algorithms mixed finite element approximations

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augmented lagrangian algorithms FOR LINEAR-PROGRAMMING

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1992年第3期75卷 445-470页

作者： GULER, O UNIV IOWA DEPT MANAGEMENT SCIIOWA CITYIA 52242

We introduce new augmented lagrangian algorithms for linear programming which provide faster global convergence rates than the augmented algorithm of Polyak and Treti'akov. Our algorithm shares the same properties as the Polyak-Treti'akov algorithm in that it terminates in finitely many iterations and obtains both primal and dual optimal solutions. We present an implementable version of the algorithm which requires only approximate minimization at each iteration. We provide a global convergence rate for this version of the algorithm and show that the primal and dual points generated by the algorithm converge to the primal and dual optimal set, respectively.

关键词： augmented lagrangian algorithms LINEAR PROGRAMMING GLOBAL CONVERGENCE RATES CONVEX PROGRAMMING

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NEW PROXIMAL POINT algorithms FOR CONVEX MINIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 1992年第4期2卷 649-664页

作者： Gueler, Osman Univ Iowa Dept Management Sci Iowa City IA 52242 USA

This paper introduces two new proximal point algorithms for minimizing a proper, lower-semicontinuous convex function f: R-n -> R boolean OR {infinity}. Under this minimal assumption on f, the first algorithm possesses the global convergence rate estimate f(x(k))-min(x is an element of Rn) f(x) = O(1/(Sigma(k-1)(j=0)root lambda(j))(2)), where {lambda(k)}(k=0)(infinity) are the proximal parameters. It is shown that this algorithm converges, and global convergence rate estimates for it are provided, even if minimizations are performed inexactly at each iteration. Both algorithms converge even if f has no minimizers or is unbounded from below. These algorithms and results are valid in infinite-dimensional Hilbert spaces.

关键词： proximal point algorithms global convergence rates augmented lagrangian algorithms convex programming

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