A class of trust region based algorithms is presented for the solution of nonlinear optimization problems with a convex feasible set. At variance with previously published analysis of this type, the theory presented allows for the use of general norms. Furthermore, the proposed algorithms do not require the explicit computation of the projected gradient, and can therefore be adapted to cases where the projection onto the feasible domain may be expensive to calculate. Strong global convergence results are derived for the class. It is also shown that the set of linear and nonlinear constraints that are binding at the solution are identified by the algorithms of the class in a finite number of iterations.

Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e. for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shiftvariant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shiftvariant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.

Nonnegative Matrix Approximation is an effective matrix decomposition technique that has proven to be useful for a wide variety of applications ranging from document analysis and image processing to bioinformatics. There exist a few algorithms for nonnegative matrix approximation (NNMA), for example, Lee & Seung’s multiplicative updates, alternating least squares, and certain gradient descent based procedures. All of these procedures suffer from either slow convergence, numerical instabilities, or at worst, theoretical unsoundness. In this paper we present new and improved algorithms for the least-squares NNMA problem, which are not only theoretically well-founded, but also overcome many of the deficiencies of other methods. In particular, we use non-diagonal gradient scaling to obtain rapid convergence. Our methods provide numerical results superior to both Lee & Seung’s method as well to the alternating least squares (ALS) heuristic, which is known to work well in some situations but has no theoretical guarantees (Berry et al. 2006). Our approach extends naturally to include regularization and box-constraints, without sacrificing convergence guarantees. We present experimental results on both synthetic and realworld datasets to demonstrate the superiority of our methods, in terms of better approximations as well as efficiency.

A survey of the development of algorithms for enforcing nonnegativity constraints in scientific computation is given. Special emphasis is placed on such constraints in least squares computations in numerical linear algebra and in nonlinear optimization. Techniques involving nonnegative low-rank matrix and tensor factorizations are also emphasized. Details are provided for some important classical and modern applications in science and engineering. For completeness, this report also includes an effort toward a literature survey of the various algorithms and applications of nonnegativity constraints in numerical analysis. Key Words: nonnegativity constraints, nonnegative least squares, matrix and tensor factorizations, image processing, optimization.

Constrained least squares estimation lies at the heart of many applications in fields as diverse as statistics, psychometrics, signal processing, or even machine learning. Nonnegativity requirements on the model variables are amongst the simplest constraints that arise naturally, and the corresponding least-squares problem is called Nonnegative Least Squares or NNLS. In this paper we present a new, efficient, and scalable Quasi-Newton-type method for solving the NNLS problem, improving on several previous approaches and leading to a superlinearly convergent method. We show experimental results comparing our method to well-known methods for solving the NNLS problem. Our method significantly outperforms other methods, especially as the problem size becomes larger. 1

this paper, we will concentrate on the overspeci#cation arising from the ASCs and will not consider other possible errors sources.

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Abstract: Nonnegative matrix approximation (NNMA) is a popular matrix decomposition technique that has proven to be useful across a diverse variety of fields with applications ranging from document analysis and image processing to bioinformatics and signal processing. Over the years, several algorithms for NNMA have been proposed, e.g. Lee and Seung’s multiplicative updates, alternating least squares (ALS), and gradient descent-based procedures. However, most of these procedures suffer from either slow convergence, numerical instability, or at worst, serious theoretical drawbacks. In this paper, we develop a new and improved algorithmic framework for the least-squares NNMA problem, which is not only theoretically well-founded, but also overcomes many deficiencies of other methods. Our framework readily admits powerful optimization techniques and as concrete realizations we present implementations based on the Newton, BFGS and conjugate gradient methods. Our algorithms provide numerical results superior to both Lee and Seung’s method as well as to the alternating least squares heuristic, which was reported to work well in some situations but has no theoretical guarantees [1]. Our approach extends naturally to include regularization and box-constraints without sacrificing convergence guarantees. We present experimental results on both synthetic and real-world datasets that demonstrate the superiority of our methods, both in terms of better approximations as well as

Preconditioning methods can accelerate the convergence of gradient-based iterative methods for tomographic image reconstruction and image restoration. Circulant preconditioners have been used extensively for shiftinvariant problems. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. For inverse problems that are approximately shiftinvariant (i.e. approximately block-Toeplitz or blockcirculant Hessians), circulant or Fourier-based preconditioners can provide remarkable acceleration. However, in applications with nonuniform noise variance (such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging), the Hessian of the (penalized) weighted least-squares objective function is quite shift-variant, and the Fourier preconditioner performs poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty ...

This paper proposes an improvement of existing methods of origin-destination matrix estimation by an explicit use of data describing the structure of the matrix. These data can be namely obtained from parking surveys. The new model is applied on both illustrative and real examples, and the results are discussed. Comparisons with the results obtained with SATURN/ME2 and the generalized least-squares method are also presented.