Jean-Pierre Dussault

Construction of Voronoi polyhedra

Abstract Given a configuration of points, a procedure for constructing the corresponding Voronoi diagram is given. The procedure is exact for molecules in the bulk. Polyhedra of surface molecules can be either eliminated or included using a periodic boundary condition. The construction is of interest in astronomy, biology, chemistry, materials science, as well as in physics (with points representing atoms, molecules, ions, etc.). The present method is more efficient than other procedures described in the literature.

A note on a globally convergent Newton method for solving monotone variational inequalities

It is well-known (see Pang and Chan [8]) that Newtons method, applied to strongly monotone variational inequalities, is locally and quadratically convergent. In this paper we show that Newtons method yields a descent direction for a non-convex, non-differentiable merit function, even in the absence of strong monotonicity. This result is then used to modify Newtons method into a globally convergent algorithm by introducing a linesearch strategy. Furthermore, under strong monotonicity (i) the optimal face is attained after a finite number of iterations, (ii) the stepsize is eventually fixed to the value one, resulting in the usual Newton step. Computational results are presented.

Numerical stability and efficiency of penalty algorithms

Penalty algorithms have been somewhat forgotten due to numerical instabilities once believed to be inherent to those methods. One usually has to solve a sequence of such problems, and when the penalty factor decreases too fast, the subproblems may become intractable. Moreover, as the penalty factor decreases, the unconstrained subproblem becomes ill conditioned, and thus difficult to solve. Also, in several intermediate computations, numerical instability may show up. The author proposes remedies to such problems and presents a wide class of numerically stable penalty algorithms. The work is done in the more general context of variational inequality problems, which encompasses optimization problems. The author’s results yield a family of globally convergent, two-step superlinearly convergent, numerically stable algorithms for variational inequality problems. Finally, issues in the numerically stable implementation of intermediate computations within those algorithms are discussed.

Convex quadratic programming with one constraint and bounded variables

In this paper we propose an iterative algorithm for solving a convex quadratic program with one equality constraint and bounded variables. At each iteration, a separable convex quadratic program with the same constraint set is solved. Two variants are analyzed: one that uses an exact line search, and the other a unit step size. Preliminary testing suggests that this approach is efficient for problems with diagonally dominant matrices.

Stable exponential-penalty algorithm with superlinear convergence

A renewed interest in penalty algorithms for solving mathematical programming problems has been motivated by some recent techniques which eliminate the ill-conditioning caused by the convergence to zero of the penalty parameter. These techniques are based on a good identification of the active set of constrainst at the optimum. In this sense, interior penalty methods to be more efficient than exterior ones, but their drawback lies in the need of an interior starting point. We propose in this paper an exponential penalty function which does not need interior starting points, but whose ultimate behavior is just like an interior penalty method. A superlinearly convergent algorithm based on the exponential penalty function is proposed.

A sequential linear programming algorithm for solving monotone variational inequalities

Applied to strongly monotone variational inequalities, Newton’s algorithm achieves local quadratic convergence. In this paper it is shown how the basic Newton method can be modified to yield an algorithm whose global convergence can be guaranteed by monitoring the monotone decrease of the “gap function” associated with the variational inequality. Each iteration consists in the solution of a linear program in the space of primal-dual variables and of a linesearch. Convergence does not depend on strong monotonicity. However, under strong monotonicity and geometric stability assumptions, the set of active constraints at the solution is implicitly identified, and quadratic convergence is achieved.

A New Regularization Scheme for Mathematical Programs with Complementarity Constraints

We propose a new regularization scheme for mathematical programs with complementarity constraints (MPCC) by relaxing all the constraints of the complementarity system. We show that, under the MPCC-linear independence constraint qualifications (MPCC-LICQ), the Lagrange multipliers exist for this regularization. Our method has strong convergence properties under MPCC-linear independence constraint qualifications and some weak conditions of the strict complementarity. In particular, under MPCC-LICQ, it is shown that any accumulation point of the regularized stationary points is M-stationary for the MPCC problem, and if the asymptotically weak nondegeneracy condition holds at a stationary point of the regularized problem, then it is strongly stationary. An algorithm for solving the proposed regularization is presented and numerical experiments are reported. Some comparisons with other methods are discussed with illustrative examples.

Optimal pulse shaping for coherent control by the penalty algorithm

Abstract We use penalty methods coupled with unitary exponential operator methods to solve the optimal control problem for molecular time-dependent Schrodinger equations involving laser pulse excitations. A stable numerical algorithm is presented which propagates directly from initial states to given final states. Results are reported for an analytically solvable model for the complete inversion of a three-state system.

A smoothing heuristic for a bilevel pricing problem

Abstract In this paper, we provide a heuristic procedure, that performs well from a global optimality point of view, for an important and difficult class of bilevel programs. The algorithm relies on an interior point approach that can be interpreted as a combination of smoothing and implicit programming techniques. Although the algorithm cannot guarantee global optimality, very good solutions can be obtained through the use of a suitable set of parameters. The algorithm has been tested on large-scale instances of a network pricing problem, an application that fits our modeling framework. Preliminary results show that on hard instances, our approach constitutes an alternative to solvers based on mixed 0–1 programming formulations.

Design of iterative ROI transmission tomography reconstruction procedures and image quality analysis

PURPOSE An iterative edge-preserving CT reconstruction algorithm for high-resolution imaging of small regions of the field of view is investigated. It belongs to a family of region-of-interest reconstruction techniques in which a low-cost pilot reconstruction of the whole field of view is first performed and then used to deduce the contribution of the region of interest to the projection data. These projections are used for a high-resolution reconstruction of the region of interest (ROI) using a regularized iterative algorithm, resulting in significant computational savings. This paper examines how the technique by which the pilot reconstruction of the full field of view is obtained affects the total runtime and the image quality in the region of interest. METHODS Previous contributions to the literature have each focused on a single approach for the pilot reconstruction. In this paper, two such approaches are compared: the filtered backprojection and a low-resolution regularized iterative reconstruction method. ROI reconstructions are compared in terms of image quality and computational cost over simulated and physical phantom (Catphan600) studies, in order to assess the compromises that most impact the quality of the ROI reconstruction. RESULTS With the simulated phantom, new artifacts that appear in the ROI images are caused by significant errors in the pilot reconstruction. These errors include excessive coarseness of the pilot image grid and beam-hardening artifacts. With the Catphan600 phantom, differences in the imaging model of the scanner and that of the iterative reconstruction algorithm cause dark border artifacts in the ROI images. CONCLUSIONS Inexpensive pilot reconstruction techniques (analytical algorithms, very-coarse-grid penalized likelihood) are practical choices in many common cases. However, they may yield background images altered by edge degradation or beam hardening, inducing projection inconsistency in the data used for ROI reconstruction. The ROI images thus have significant streak and speckle artifacts, which adversely affect the resolution-to-noise compromise. In these cases, edge-preserving penalized-likelihood methods on not-too-coarse image grids prove to be more robust and provide the best ROI image quality.