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تسجيل مستخدم جديدProjection Methods: An Annotated Bibliography of Books and Reviews
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Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present then projections (or approximate projections) onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given family of individual sets. Projection methods employ projections (or approximate projections) onto convex sets in various ways. They may use different kinds of projections and, sometimes, even use different projections within the same algorithm. They serve to solve a variety of problems which are either of the feasibility or the optimization types. They have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavior patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological, and mathematical problems. This annotated bibliography includes books and review papers on, or related to, projection methods that we know about, use, and like. If you know of books or review papers that should be added to this list please contact us.
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The decomposition of a matrix, as a product of factors with particular properties, is a much used tool in numerical analysis. Here we develop methods for decomposing a matrix $C$ into a product $X Y$, where the factors $X$ and $Y$ are required to min
imize their distance from an arbitrary pair $X_0$ and $Y_0$. This type of decomposition, a projection to a matrix product constraint, in combination with projections that impose structural properties on $X$ and $Y$, forms the basis of a general method of decomposing a matrix into factors with specified properties. Results are presented for the application of these methods to a number of hard problems in exact factorization.
We revisit the feasibility approach to the construction of compactly supported smooth orthogonal wavelets on the line. We highlight its flexibility and illustrate how symmetry and cardinality properties are easily embedded in the design criteria. We
solve the resulting wavelet feasibility problems using recently introduced centering methods, and we compare performance. Solutions admit real-valued compactly supported smooth orthogonal scaling functions and wavelets with near symmetry and near cardinality properties.
Small-scale Mixed-Integer Quadratic Programming (MIQP) problems often arise in embedded control and estimation applications. Driven by the need for algorithmic simplicity to target computing platforms with limited memory and computing resources, this
paper proposes a few approaches to solving MIQPs, either to optimality or suboptimally. We specialize an existing Accelerated Dual Gradient Projection (GPAD) algorithm to effectively solve the Quadratic Programming (QP) relaxation that arise during Branch and Bound (B&B) and propose a generic framework to warm-start the binary variables which reduces the number of QP relaxations. Moreover, in order to find an integer feasible combination of the binary variables upfront, two heuristic approaches are presented: ($i$) without using B&B, and ($ii$) using B&B with a significantly reduced number of QP relaxations. Both heuristic approaches return an integer feasible solution that may be suboptimal but involve a much reduced computation effort. Such a feasible solution can be either implemented directly or used to set an initial upper bound on the optimal cost in B&B. Through different hybrid control and estimation examples involving binary decision variables, we show that the performance of the proposed methods, although very simple to code, is comparable to that of state-of-the-art MIQP solvers.
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