Abstract. The Time-Frequency and Time-Scale communities have recently developed a large numberofovercompletewaveform dictionaries| stationarywavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition i n to overcomplete systems is not unique, and several methods for decomposition have been proposed, including the Method of Frames (MOF), Matching Pursuit (MP), and, for special dictionaries, the Best Orthogonal Basis (BOB).Basis Pursuit (BP) is a principle for decomposing a signal into an \optimal" superposition of dictionary elements, where optimal means having the smallest l 1 norm of coecients among all such decompositions. We give examples exhibiting several advantages over MOF, MP and BOB, including better sparsity, and super-resolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation de-noising, and multi-scale edge de-noising.Basis Pursuit in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by i n terior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

Abstract. The Time-Frequency and Time-Scale communities have recently developed a large numberofovercompletewaveform dictionaries| stationarywavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition i n to overcomplete systems is not unique, and several methods for decomposition have been proposed, including the Method of Frames (MOF), Matching Pursuit (MP), and, for special dictionaries, the Best Orthogonal Basis (BOB).Basis Pursuit (BP) is a principle for decomposing a signal into an \optimal" superposition of dictionary elements, where optimal means having the smallest l 1 norm of coecients among all such decompositions. We give examples exhibiting several advantages over MOF, MP and BOB, including better sparsity, and super-resolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation de-noising, and multi-scale edge de-noising.Basis Pursuit in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by i n terior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

An iterative method is given for solving Ax ~ffi b and minU Ax -b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties.Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method, subroutine LSQR. Numerical tests are described comparing I~QR with several other conjugate-gradient algorithms, indicating that I~QR is the most reliable algorithm when A is ill-conditioned.

Sequential quadratic programming (SQP) methods have proved highly effective for solving constrained optimization problems with smooth nonlinear functions in the objective and constraints. Here we consider problems with general inequality constraints (linear and nonlinear). We assume that first derivatives are available and that the constraint gradients are sparse. Second derivatives are assumed to be unavailable or too expensive to calculate.We discuss an SQP algorithm that uses a smooth augmented Lagrangian merit function and makes explicit provision for infeasibility in the original problem and the QP subproblems. The Hessian of the Lagrangian is approximated using a limited-memory quasi-Newton method.SNOPT is a particular implementation that uses a reduced-Hessian semidefinite QP solver (SQOPT) for the QP subproblems. It is designed for problems with many thousands of constraints and variables but is best suited for problems with a moderate number of degrees of freedom (say, up to 2000). Numerical results are given for most of the CUTEr and COPS test collections (about 1020 examples of all sizes up to 40000 constraints and variables, and up to 20000 degrees of freedom). become large), SNOPT enters nonlinear elastic mode and solves the problemis called a composite objective, and the penalty parameter γ (γ ≥ 0) may take a finite sequence of increasing values. If (NP) has a feasible solution and γ is sufficiently large, the solutions to (NP) and (NP(γ)) are identical. If (NP) has no feasible solution, (NP(γ)) will tend to determine a "good" infeasible point if γ is again sufficiently large. (If γ were infinite, the nonlinear constraint violations would be minimized subject to the linear constraints and bounds.) A similar 1 formulation of (NP) is used in the SQP method of Tone [98] and is fundamental to the S 1 QP algorithm of Fletcher [38]. See also Conn [25] and Spellucci [94]. An attractive feature is that only linear terms are added to (NP), giving no increase in the expected degrees of freedom at each QP solution. 1.2. The SQP Approach. An SQP method was first suggested by Wilson [102] for the special case of convex optimization. The approach was popularized mainly by Biggs [7], Han [66], and Powell [85, 87] for general nonlinear constraints. Further history of SQP methods and extensive bibliographies are given in [61, 39, 73, 78, 28]. For a survey of recent results, see Gould and Toint [65]. Several general-purpose SQP solvers have proved reliable and efficient during the last 20 years. For example, under mild conditions the solvers NLPQL [92], NPSOL [57, 60], and DONLP [95] typically find a (local) optimum from an arbitrary starting point, and they require relatively few evaluations of the problem functions and gradients compared to traditional solvers such as MINOS [75, 76, 77] and CONOPT [34, 2].SQP methods have been particularly successful in solving the optimization problems arising in optimal trajectory calculations. For many years, the optimal trajectory system OTIS (Hargraves and Paris [67]) has been...

COnstraint-Based Reconstruction and Analysis (COBRA) provides a molecular mechanistic framework for integrative analysis of experimental data and quantitative prediction of physicochemically and biochemically feasible phenotypic states. The COBRA Toolbox is a comprehensive software suite of interoperable COBRA methods. It has found widespread applications in biology, biomedicine, and biotechnology because its functions can be flexibly combined to implement tailored COBRA protocols for any biochemical network. Version 3.0 includes new methods for quality controlled reconstruction, modelling, topological analysis, strain and experimental design, network visualisation as well as network integration of chemoinformatic, metabolomic, transcriptomic, proteomic, and thermochemical data. New multi-lingual code integration also enables an expansion in COBRA application scope via high-precision, high-performance, and nonlinear numerical optimisation solvers for multi-scale, multi-cellular and reaction kinetic modelling, respectively. This protocol can be adapted for the generation and analysis of a constraint-based model in a wide variety of molecular systems biology scenarios. This protocol is an update to the COBRA Toolbox 1.0 and 2.0. The COBRA Toolbox 3.0 provides an unparalleled depth of constraint-based reconstruction and analysis methods. ]); 61 | The MUST sets are the sets of reactions that must increase or decrease their flux in order to achieve the desired phenotype in the mutant strain. As shown in Figure 6, the first order MUST sets are MustU and MustL while second order MUST sets are denoted as MustUU, MustLL, and MustUL. After parameters and constraints are defined, the functions findMustL and findMustU are run to determine the mustU and mustL sets, respectively. Define an ID of the run with:Each time the MUST sets are determined, folders are generated to read inputs and store outputs, i.e., reports. These folders are located in the directory defined by the uniquely defined runID.62 | In order to find the first order MUST sets, constraints should be defined: >> constrOpt = struct('rxnList', {{'EX_gluc', 'R75', 'EX_suc'}}, 'values', [-100; 0; 155.5]); 63 | The first order MUST set MustL is determined by running: >> [mustLSet, pos_mustL] = findMustL(model, minFluxesW, maxFluxesW, ... 'constrOpt', constrOpt, 'runID', runID);If runID is set to 'TestoptForceL', a folder TestoptForceL is created, in which two additional folders InputsMustL and OutputsMustL are created. The InputsMustL folder contains all the inputs required to run the function findMustL, while the OutputsMustL folder contains the mustL set found and a report that summarises all the inputs and outputs. In order to maintain a chronological order of computational experiments, the report is timestamped.64 | Display the reactions that belong to the mustL set using: >> disp(mustLSet) 65 | The first order MUST set MustU is determined by running: >> [mustUSet, pos_mustU] = findMustU(model, minFluxesW, maxFluxesW, ... 'constrOpt', constrOpt, 'runID', runID);...