William Hager

• University of Florida

For polyhedral constrained optimization problems and a feasible point $$\textbf{x}$$ x , it is shown that the projection of the negative gradient on the tangent cone, denoted $$\nabla _\varOmega f(\textbf{x})$$ ∇ Ω f ( x ) , has an orthogonal decomposition of the form $$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$ β ( x ) + φ ( x )...

The Polyhedral Active Set Algorithm (PASA) is designed to optimize a general nonlinear function over a polyhedron. Phase one of the algorithm is a nonmonotone gradient projection algorithm, while phase two is an active set algorithm that explores faces of the constraint polyhedron. A gradient-based implementation is presented, where a projected ver...

An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Pro...

The Polyhedral Active Set Algorithm (PASA) is designed to optimize a general nonlinear function over a polyhedron. Phase one of the algorithm is a nonmonotone gradient projection algorithm, while phase two is an active set algorithm that explores faces of the constraint polyhedron. A gradient-based implementation is presented, where a projected ver...

A mesh refinement method is described for solving optimal control problems using Legendre‐Gauss‐Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targeted h‐refinement approach wher...

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a...

Convergence rates are established for an inexact accelerated alternating direction method of multipliers (I-ADMM) for general separable convex optimization with a linear constraint. Both ergodic and non-ergodic iterates are analyzed. Relative to the iteration number k, the convergence rate is \(\mathcal{{O}}(1/k)\) in a convex setting and \(\mathca...

A mesh refinement method is described for solving optimal control problems using Legendre-Gauss-Radau collocation. The method detects discontinuities in the control solution by employing an edge detection scheme based on jump function approximations. When discontinuities are identified, the mesh is refined with a targeted $h$-refinement approach wh...

Convergence rates are established for an inexact accelerated alternating direction method of multipliers (I-ADMM) for general separable convex optimization with a linear constraint. Both ergodic and non-ergodic iterates are analyzed. Relative to the iteration number k, the convergence rate is O(1/k) in a convex setting and O(1/k^2) in a strongly co...

A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre-Gauss-Radau orthogonal direct collocation method. This modified Legendre-Gauss-Radau method adds two variables and two constraints at the end of a mesh interval when compared with a...

For unconstrained control problems, a local convergence rate is established for an $hp$-orthogonal collocation method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discr...

For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem po...

A mesh refinement method is developed for solving bang-bang optimal control problems using direct collocation. The method starts by finding a solution on a coarse mesh. Using this initial solution, the method then determines automatically if the Hamiltonian is linear with respect to the control, and, if so, estimates the locations of the discontinu...

A method is described for computational optimal guidance and control using adaptive Gaussian quadrature collocation and sparse nonlinear programming. The method employs adaptive Legendre-Gauss-Radau (LGR) quadrature collocation using a mesh truncation and remapping procedure at the start of each guidance cycle, thereby retaining only the mesh point...

Inexact alternating direction multiplier methods (ADMMs) are developed for solving general separable convex optimization problems with a linear constraint and with an objective that is the sum of smooth and nonsmooth terms. The approach involves linearized subproblems, a back substitution step, and either gradient or accelerated gradient techniques...

An adaptive mesh refinement method for solving optimal control problems is described. The method employs orthogonal collocation at Legendre-Gauss-Radau points. Accuracy in the method is achieved by adjusting the number of mesh intervals, the polynomial degree within each mesh interval, and, when possible, reducing the mesh size. The decision to inc...

A local convergence rate is established for a Gauss orthogonal collocation method applied to optimal control problems with control constraints. If the Hamiltonian possesses a strong convexity property, then the theory yields convergence for problems whose optimal state and costate possess two square integrable derivatives. The convergence theory is...

A local convergence rate is established for a Gauss orthogonal collocation method applied to optimal control problems with control constraints. If the Hamiltonian possesses a strong convexity property, then the theory yields convergence for problems whose optimal state and costate possess two square integrable derivatives. The convergence theory is...

A polyhedral active set algorithm PASA is developed for solving a nonlinear optimization problem whose feasible set is a polyhedron. Phase one of the algorithm is the gradient projection method, while phase two is any algorithm for solving a linearly constrained optimization problem. Rules are provided for branching between the two phases. Global c...

A polyhedral active set algorithm PASA is developed for solving a nonlinear optimization problem whose feasible set is a polyhedron. Phase one of the algorithm is the gradient projection method, while phase two is any algorithm for solving a linearly constrained optimization problem. Rules are provided for branching between the two phases. Global c...

A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood o...

For unconstrained control problems, a local convergence rate is established for an $hp$-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a...

Inexact alternating direction multiplier methods (ADMMs) are developed for solving general separable convex optimization problems with a linear constraint and with an objective that is the sum of smooth and nonsmooth terms. The approach involves linearized subproblems, a back substitution step, and either gradient or accelerated gradient techniques...

An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing φ(Bu) + H(u), where H and φ are convex functions, and φ is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance image reconstruction problems. In this pape...

Estimates are obtained for the Lebesgue constants associated with the Gauss quadrature points on $(-1, +1)$ augmented by the point $-1$ and with the Radau quadrature points on either $(-1, +1]$ or $[-1, +1)$. It is shown that the Lebesgue constants are $O(\sqrt{N})$, where $N$ is the number of quadrature points. These point sets arise in the estima...

An earlier paper proved the convergence of a variable stepsize Bregman operator splitting algorithm (BOSVS) for minimizing Φ(u, w) = φ(w) + H(u) subject to a constraint w = Bu, where H and φ are convex functions, and φ is possibly nonsmooth. The algorithm was shown to be relatively efficient when applied to partially parallel magnetic resonance ima...

Recently an affine scaling, interior point algorithm ASL was developed for box constrained optimization problems with a single linear constraint (Gonzalez-Lima et al., SIAM J. Optim. 21:361–390, 2011). This note extends the algorithm to handle more general polyhedral constraints. With a line search, the resulting algorithm ASP maintains the global...

An alternating direction approximate Newton method (ADAN) is developed for solving inverse problems of the form min{φ(Bu) + (1/2) − f 2 2 }, where φ is convex and possibly nonsmooth, and A and B are matrices. Problems of this form arise in image reconstruction where A is the matrix describing the imaging device, f is the measured data, φ is a regul...

Two methods are presented for approximating the costate of optimal control problems in integral form using orthogonal collocation at Legendre–Gauss (LG) and Legendre–Gauss–Radau (LGR) points. It is shown that the derivative of the costate of the continuous-time optimal control problem is equal to the negative of the costate of the integral form of...

SUMMARYA mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the differ...

A method is presented for costate estimation of state-inequality constrained optimal control problems using orthogonal collocation at Legendre-Gauss-Radau points. It is shown that the Lagrange multipliers of the nonlinear programming problem can be accurately mapped to the costates of the continuous-time optimal control problem. The differentiation...

We propose a new gradient method for quadratic programming, named SDC, which alter-nates some SD iterates with some gradient iterates that use a constant steplength computed through the Yuan formula. The SDC method exploits the asymptotic spectral behaviour of the Yuan steplength to foster a selective elimination of the components of the gradient a...

In measurements of the electric field associated with the current of a sprite 450 km from ground-based field sensors, it was observed that the sign of the electric field was positive when positive charge was lowered from the ionosphere. A recent model for the electric field associated with the sprite current also predicts positive field changes at...

This paper develops a Bregman operator splitting algorithm with variable stepsize (BOSVS) for solving problems of the form $\min\{\phi(Bu) +1/2\|Au-f\|_{2}^{2}\}$ , where ϕ may be nonsmooth. The original Bregman Operator Splitting (BOS) algorithm employed a fixed stepsize, while BOSVS uses a line search to achieve better efficiency. These schemes...

In theory, the successive gradients generated by the conjugate gradient method applied to a quadratic should be orthogonal. However, for some ill-conditioned problems, orthogonality is quickly lost due to rounding errors, and convergence is much slower than expected. A limited memory version of the nonlinear conjugate gradient method is developed....

Charge rearrangement by sprites is analyzed for a mesoscale convective system (MCS) situated in north Texas and east New Mexico on 15 July 2010. During the thunderstorm, electric field data were recorded by the Langmuir Electric Field Array (LEFA), while magnetic field data were recorded by the charge-moment network near Duke University. A high spe...

A new algorithm is presented for efficiently solving image reconstruction problems that arise in partially parallel magnetic resonance imaging. This algorithm minimizes an objective function of the form φ(Bu) + 1/2||FpSu - f||2, where φ is the regularization term which may be nonsmooth. In image reconstruction, the φ term corresponds to total varia...

This paper presents two fast algorithms for total variation-based image reconstruction in a magnetic resonance imaging technique known as partially parallel imaging (PPI), where the inversion matrix is large and ill-conditioned. These algorithms utilize variable splitting techniques to decouple the original problem into more easily solved subproble...

An hp-adaptive pseudospectral method is presented for numerically solving optimal control problems. The method presented in this paper iteratively determines the number of segments, the width of each segment, and the polynomial degree required in each segment in order to obtain a solution to a user-specified accuracy. Starting with a global pseudos...

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates...

A variable-order adaptive pseudospectral method is presented for solving optimal control problems. The method developed in this paper adjusts both the mesh spacing and the degree of the polynomial on each mesh interval until a specified error tolerance is satisfied. In regions of relatively high curvature, convergence is achieved by refining the me...

An important aspect of numerically approximating the solution of an infinite-horizon optimal control problem is the manner in which the horizon is treated. Generally, an infinite-horizon optimal control problem is approximated with a finite-horizon problem. In such cases, regardless of the finite duration of the approximation, the final time lies a...

An affine-scaling algorithm (ASL) for optimization problems with a single linear equality constraint and box restrictions is developed. The algorithm has the property that each iterate lies in the relative interior of the feasible set. The search direction is obtained by approximating the Hessian of the objective function in Newton's method by a mu...

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre–Gauss (LG), Legendre–Gauss–Radau (LGR), and Legendre–Gauss–Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. C...

Lightning charge transport is analyzed for a thunderstorm which occurred on 18 August 2004 near Langmuir Laboratory in New Mexico. The analysis employs wide band measurements of the electric field by a balloon-borne electric field sonde or Esonde, simultaneous Lightning Mapping Array measurements of VHF pulses emitted during lightning breakdown, an...

We develop an affine-scaling algorithm for box-constrained optimization which has the property that each iterate is a scaled cyclic Barzilai–Borwein (CBB) gradient iterate that lies in the interior of the feasible set. Global convergence is established for a nonmonotone line search, while there is local R-linear convergence at a nondegenerate local...

The supernodal method for sparse Cholesky factorization represents the factor L as a set of supernodes, each consisting of a contiguous set of columns of L with identical nonzero pattern. A conventional supernode is stored as a dense submatrix. While this is suitable for sparse Cholesky factorization where the nonzero pattern of L does not change,...

An overview is presented of three different pseudospectral methods based on collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. For each of the schemes presented here, (1) the state at the final time can be expressed in terms of a quadrature rule associated with the collocatio n points, (2) the s...

CHOLMOD is a set of routines for factorizing sparse symmetric positive definite matrices of the form A or AA T , updating/downdating a sparse Cholesky factorization, solving linear systems, updating/downdating the solution to the triangular system Lx = b , and many other sparse matrix functions for both symmetric and unsymmetric matrices. Its super...

The following generalized eigenproblem is analyzed: Find u∈H01(Ω), u≠0, and λ∈R such that〈∇u,∇v〉D=λ〈∇u,∇v〉Ω for all v∈H01(Ω), where Ω⊂Rn is a bounded domain, D is a subdomain with closure contained in Ω, and 〈⋅,⋅〉Ω is the inner product〈∇u,∇v〉Ω=∫Ω∇u⋅∇vdx. It is proved that any f∈H01(Ω) can be expanded in terms of orthogonal eigenfunctions for the ge...

Given a complex matrix H, we consider the decomposition H = QRP*, where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diago...

We present an implementation of the LP Dual Active Set Algorithm (LP DASA) based on a quadratic proximal approximation, a strategy for dropping inactive equations from the constraints, and recently developed algorithms for updating a sparse Cholesky factorization after a low-rank change. Although our main focus is linear programming, the first and...

We study the structure of dual optimization problems associated with linear constraints, bounds on the variables, and separable cost. We show how the separability of the dual cost function is related to the sparsity structure of the linear equations. As a result, techniques for ordering sparse matrices based on nested dissection or graph partitioni...

We propose a class of self-adaptive proximal point methods suitable for degenerate optimization problems where multiple minimizers may exist, or where the Hessian may be singular at a local minimizer. If the proximal regularization parameter has the form m(x)=b||Ñf(x)||h\mu({\bf{x}})=\beta\|\nabla f({\bf{x}})\|^{\eta} where η∈[0,2) and β>0 is a c...

CHOLMOD is a set of routines for factorizing sparse symmetric positive definite matrices of the form A or AAT, updating/downdating a sparse Cholesky factorization, solving linear systems, updating/downdating the solution to the triangular system Lx = b, and many other sparse matrix functions for both symmetric and unsymmetric matrices. Its supernod...

A class of least squares problems that arises in linear Bayesian estimation is analyzed. The data vector ${\bf y}$ is given by the model ${\bf y} = {\bf P}({\bf H}\bm{\theta} + \bm{\eta}) + {\bf w}$, where ${\bf H}$ is a known matrix, while $\bm{\theta}$, $\bm{\eta}$, and ${\bf w}$ are uncorrelated random vectors. The goal is to obtain the best est...

We describe a domain decomposition approach applied to the specific context of electronic structure calculations. The approach has been introduced in [BCHL07]. We survey here the computational context, and explain the peculiarities of the approach as compared to problems of seemingly the same type in other engineering sciences. Improvements of the...

We develop a computationally efficient approximation of the maximum likelihood (ML) detector for 16 quadrature amplitude modulation (16-QAM) in multiple-input multiple-output (MIMO) systems. The detector is based on solving a convex relaxation of the ML problem by an affine-scaling cyclic Barzila-Borwein method for box constrained optimization. Sim...

A model for a viscoelastic coating is derived and the coating reflectivity is determined. A coating which minimizes the reflection of sound waves is partly bang-bang and partly singular. A new gradient type algorithm is presented for computing the optimal control. Formulae for the gradient of the cost in optimal control are derived in various setti...

Techniques are developed for processing the wide band measurements of electric field obtained by a balloon-borne electric field sonde (or Esonde), and for estimating the charge transport associated with lightning. The techniques use Lightning Mapping Array measurements of the VHF pulses generated during lightning recorded simultaneously with the Es...

In this paper, we study the problem of estimating correlated multiple-input multiple-output (MIMO) channels in the presence of colored interference. The linear minimum mean square error (MMSE) channel estimator is derived and the optimal training sequences are designed based on the MSE of channel estimation. We propose an algorithm to estimate the...

Finite dimensional local convergence results for self-adaptive proximal point methods and nonlinear functions with multiple minimizers are generalized and extended to a Hilbert space setting. The principle assumption is a local error bound condition which relates the growth in the function to the distance to the set of minimizers. A local convergen...

We describe a domain decomposition approach applied to the specific context of electronic structure calculations. The approach has been introduced in Ref 1. We briefly present the algorithm and its parallel implementation. Simulation results on linear hydrocarbon chains with up to 2 millions carbon atoms are reported. Linear scaling with respect to...

We develop a computationally efficient approximation of the maximum likelihood (ML) de- tector for 16 quadrature amplitude modulation (16-QAM) in multiple-input multiple-output (MIMO) systems. The detector is based on solving a convex relaxation of the ML problem by a box constrained optimization scheme. Simulation results in a random MIMO system s...

Two matrix optimization problems are analyzed. These problems arise in signal processing and commu-nication. In the first problem, the trace of the mean square error matrix is minimized, subject to a power constraint. The solution is the training sequence, which yields the best estimate of a communication channel. The solution is expressed in terms...

Recently, a new nonlinear conjugate gradient scheme was developed which satisfies the descent condition g T k d k ≤ −7/8 V g k V 2 and which is globally convergent whenever the line search fulfills the Wolfe conditions. This article studies the convergence behavior of the algorithm; extensive numerical tests and comparisons with other methods for l...

Formulas are derived for the reflection and transmission tensors associated with a plane elastic wave impinging obliquely upon a stratified slab interposed between two homogeneous half-spaces. Both solid-solid and solid-liquid interfaces are considered.

An active set algorithm (ASA) for box constrained optimization is developed. The algorithm consists ofa nonmonotone gradient projection step, an unconstrained optimization step, and a set ofrules f or branching between the two steps. Global convergence to a stationary point is established. For a nondegenerate stationary point, the algorithm eventua...

Recently, a new nonlinear conjugate gradient scheme was developed which satisfies the descent condition gTkdk ≤ −7/8 ‖gk‖2 and which is globally convergent whenever the line search fulfills the Wolfe conditions. This article studies the convergence behavior of the algorithm; extensive numerical tests and comparisons with other methods...

This paper reviews the development of different versions of nonlinear conjugate gradient methods, with special attention given to global convergence properties.

In the cyclic Barzilai–Borwein (CBB) method, the same Barzilai–Borwein (BB) stepsize is reused for m consecutive iterations. It is proved that CBB is locally linearly convergent at a local minimizer with positive definite Hessian. Numerical evidence indicates that when m > n/2 ≥ 3, where n is the problem dimension, CBB is locally superlinearly conv...

In an earlier paper (Minimizing a quadratic over a sphere ,S IAM J. Optim., 12 (2001), 188-208), we presented the sequential subspace method (SSM) for minimizing a quadratic over a sphere. This method generates ap- proximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is compu...

Given a complex matrix H, we consider the decomposition H = QRP* where Q and P have orthonormal columns, and R is a real upper triangular matrix with diagonal elements equal to the geometric mean of the positive singular values of H. This decomposition, which we call the geometric mean decomposition, has application to signal processing and to the...

A new nonlinear conjugate gradient method and an associated implementation, based on an inexact line search, are proposed and analyzed. With exact line search, our method reduces to a nonlinear version of the Hestenes-Stiefel conjugate gradient scheme. For any (inexact) line search, our scheme satisfies the descent condition g kTd k ≤ -7/8||gk|| 2....

Given a sparse, symmetric positive definite matrix C and an associated sparse Cholesky factorization LDL,, we develop sparse techniques for updating the factorization after a symmetric modification of a row and column of C. We show how the modification in the Cholesky factorization associated with this rank two modification of C can be computed eci...

The gradient projection algorithm for function minimization is often implemented using an approximate local minimization along the projected negative gradient. On the other hand, for some difficult combinational optimization problems, where a starting guess may be far from a solution, it may be advantageous to perform a nonlocal (exact) line search...

In a seminal paper (An ecient heuristic procedure for partitioning graphs, Bell System Technical Journal, 49 (1970), pp. 291-307), Kernighan and Lin propose a pair exchange algorithm for approximating the solution to min-cut graph partitioning problems. In their algorithm, a vertex from one set in the current partition is exchanged with a vertex in...

A new nonmonotone line search algorithm is proposed and analyzed. In our scheme, we require that an average of the successive function values decreases, while the traditional nonmono- tone approach of Grippo, Lampariello, and Lucidi (SIAM J. Numer. Anal., 23 (1986), pp. 707-716) requires that a maximum of recent function values decreases. We prove...

First Page of the Article

S. M. Robinson proved several implicit function theorems which have played an important role in the development of set-valued analysis and stability analysis in optimization. We show that Robinson’s theorems can be obtained from a single implicit function-type result for set-valued maps based on a refinement of the Nadler’s fixed point theorem. A c...

The Dual Active Set Algorithm (DASA), presented in Hager, Advances in Optimization and Parallel Computing, P.M. Pardalos (Ed.), North Holland: Amsterdam, 1992, pp. 137–142, for strictly convex optimization problems, is extended to handle linear programming problems. Line search versions of both the DASA and the LPDASA are given.

Local optimality conditions are given for a quadratic programming formulation of the multiset graph partitioning problem. These conditions are related to the structure of the graph and properties of the weights.

We analyze the Euler approximation to a state constrained control problem. We show that if the active constraints satisfy an independence condition and the Lagrangian satisfies a coercivity condition, then locally there exists a solution to the Euler discretization, and the error is bounded by a constant times the mesh size. The proof couples recen...

In a seminal paper (An ecient heuristic procedure for partitioning graphs, Bell System Technical Journal, 49 (1970), pp. 291-307), Kernighan and Lin propose a pair exchange algorithm for approximating the solution to min-cut graph partitioning problems. In their algorithm, a vertex from one set in the current partition is exchanged with a vertex in...

In an abstract framework, we study local convergence properties of Newton's method for a sequence of generalized equations which models a discretized variational inequality. We identify conditions under which the method is locally quadratically convergent, uniformly in the discretization. Moreover, we show that the distance between the Newton seque...

In this paper, we analyze second-order Runge-Kutta approximations to a nonlinear optimal control problem with control constraints. If the optimal control has a derivative of bounded variation and a coercivity condition holds, we show that for a special class of Runge-Kutta schemes, the error in the discrete approximating control is O(h ) where h is...

. Given a sparse symmetric positive definite matrix AA T and an associated sparse Cholesky factorization LDL T or LL T , we develop sparse techniques for obtaining the new factorization associated with either adding a column to A or deleting a column from A. Our techniques are based on an analysis and manipulation of the underlying graph structure...

A continuous quadratic programming formulation is given for min-cut graph partitioning problems. In these problems, we partition the vertices of a graph into a collection of disjoint sets satisfying specified size constraints, while minimizing the sum of weights of edges connecting vertices in different sets. An optimal solution is related to an ei...

A quadratic programming approach is described for solving the graph partitioning problem, in which the optimal solution to a continuous problem is the exact solution to the discrete graph partitioning problem. We discuss techniques for approximating the solution to the quadratic program using gradient projections, preconditioned conjugate gradients...

An algorithm presented in [11] for diagonalizing a matrix is generalized to a block matrix setting. It is shown that the resulting algorithm is locally quadratically convergent. A global convergence proof is given for matrices with separated eigenvalues and with relatively small off-diagonal elements. Numerical examples along with comparisons to th...

. An overview is given for a new algorithm, the LP Dual Active Set Algorithm, to solve linear programming problems. In its pure form, the algorithm uses a series of projections to ascend the dual function. These projections can be approximated using proximal techniques, and both iterative and direct methods can be applied to obtain highly accurate,...