Generalized Newton method for linear optimization problems with inequality constraints

A dual problem of linear programming is reduced to the unconstrained maximization of a concave piecewise quadratic function for sufficiently large values of a certain parameter. An estimate is given for the threshold value of the parameter starting from which the projection of a given point to the set of solutions of the dual linear programming problem in dual and auxiliary variables is easily found by means of a single solution of the unconstrained maximization problem. The unconstrained maximization is carried out by the generalized Newton method, which is globally convergent in an a finite number of steps. The results of numerical experiments are presented for randomly generated large-scale linear programming problems.     

A matrix-free implementation of Riemannian Newton’s method on the Stiefel manifold

Newton’s method for unconstrained optimization problems on the Euclidean space can be generalized to that on Riemannian manifolds. The truncated singular value problem is one particular problem defined on the product of two Stiefel manifolds, and an algorithm of the Riemannian Newton’s method for this problem has been designed. However, this algorithm is not easy to implement in its original form because the Newton equation is expressed by a system of matrix equations which is difficult to solve directly. In the present paper, we propose an effective implementation of the Newton algorithm. A matrix-free Krylov subspace method is used to solve a symmetric linear system into which the Newton equation is rewritten. The presented approach can be used on other problems as well. Numerical experiments demonstrate that the proposed method is effective for the above optimization problem.     

Global convergence and the Powell singular function

The Powell singular function was introduced 1962 by M.J.D. Powell as an unconstrained optimization problem. The function is also used as nonlinear least squares problem and system of nonlinear equations. The function is a classic test function included in collections of test problems in optimization as well as an example problem in text books. In the global optimization literature the function is stated as a difficult test case. The function is convex and the Hessian has a double singularity at the solution. In this paper we consider Newton’s method and methods in Halley class and we discuss the relationship between these methods on the Powell Singular Function. We show that these methods have global but linear rate of convergence. The function is in a subclass of unary functions and results for Newton’s method and methods in the Halley class can be extended to this class. Newton’s method is often made globally convergent by introducing a line search. We show that a full Newton step will satisfy many of standard step length rules and that exact line searches will yield slightly faster linear rate of convergence than Newton’s method. We illustrate some of these properties with numerical experiments.     

Distance majorization and its applications

The problem of minimizing a continuously differentiable convex function over an intersection of closed convex sets is ubiquitous in applied mathematics. It is particularly interesting when it is easy to project onto each separate set, but nontrivial to project onto their intersection. Algorithms based on Newton’s method such as the interior point method are viable for small to medium-scale problems. However, modern applications in statistics, engineering, and machine learning are posing problems with potentially tens of thousands of parameters or more. We revisit this convex programming problem and propose an algorithm that scales well with dimensionality. Our proposal is an instance of a sequential unconstrained minimization technique and revolves around three ideas: the majorization-minimization principle, the classical penalty method for constrained optimization, and quasi-Newton acceleration of fixed-point algorithms. The performance of our distance majorization algorithms is illustrated in several applications.     

A smoothing Newton method for a type of inverse semi-definite quadratic programming problem

We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC¹${\text{SC}}^1$) convex programming problem with fewer variables than the original one. The Karush–Kuhn–Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush–Kuhn–Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.     

A feasible semismooth asymptotically Newton method for mixed complementarity problems

 Semismooth Newton methods constitute a major research area for solving mixed complementarity problems (MCPs). Early research on semismooth Newton methods is mainly on infeasible methods. However, some MCPs are not well defined outside the feasible region or the equivalent unconstrained reformulations of other MCPs contain local minimizers outside the feasible region. As both these problems could make the corresponding infeasible methods fail, more recent attention is on feasible methods. In this paper we propose a new feasible semismooth method for MCPs, in which the search direction asymptotically converges to the Newton direction. The new method overcomes the possible non-convergence of the projected semismooth Newton method, which is widely used in various numerical implementations, by minimizing a one-dimensional quadratic convex problem prior to doing (curved) line searches. As with other semismooth Newton methods, the proposed method only solves one linear system of equations at each iteration. The sparsity of the Jacobian of the reformulated system can be exploited, often reducing the size of the system that must be solved. The reason for this is that the projection onto the feasible set increases the likelihood of components of iterates being active. The global and superlinear/quadratic convergence of the proposed method is proved under mild conditions. Numerical results are reported on all problems from the MCPLIB collection [8]. Received: December 1999 / Accepted: March 2002 Published online: September 5, 2002 RID="★" ID="★" This work was supported in part by the Australian Research Council. Key Words. mixed complementarity problems – semismooth equations – projected Newton method – convergence AMS subject classifications. 90C33, 90C30, 65H10     

A Perturbation approach for an inverse quadratic programming problem

We consider an inverse quadratic programming (QP) problem in which the parameters in both the objective function and the constraint set of a given QP problem need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a linear complementarity constrained minimization problem with a positive semidefinite cone constraint. With the help of duality theory, we reformulate this problem as a linear complementarity constrained semismoothly differentiable (SC¹) optimization problem with fewer variables than the original one. We propose a perturbation approach to solve the reformulated problem and demonstrate its global convergence. An inexact Newton method is constructed to solve the perturbed problem and its global convergence and local quadratic convergence rate are shown. As the objective function of the problem is a SC¹ function involving the projection operator onto the cone of positively semi-definite symmetric matrices, the analysis requires an implicit function theorem for semismooth functions as well as properties of the projection operator in the symmetric-matrix space. Since an approximate proximal point is required in the inexact Newton method, we also give a Newton method to obtain it. Finally we report our numerical results showing that the proposed approach is quite effective.     

Quadratic convergence of a smoothing Newton method for symmetric cone programming without strict complementarity

This paper deals with symmetric cone programming (SCP), which includes the linear programming (LP), the second-order cone programming (SOCP), the semidefinite programming (SDP) as special cases. Based on the Chen?CMangasarian smoothing function of the projection operator onto symmetric cones, we establish a smoothing Newton method for SCP. Global and quadratic convergence of the proposed algorithm is established under the primal and dual constraint nondegeneracies, but without the strict complementarity. Moreover, we show the equivalence at a Karush?CKuhn?CTucker (KKT) point among the primal and dual constraint nondegeneracies, the nonsingularity of the B-subdifferential of the smoothing counterpart of the KKT system, and the nonsingularity of the corresponding Clarke??s generalized Jacobian.     

A half-space projection method for solving generalized Nash equilibrium problems

The generalized Nash equilibrium problem (GNEP) is an n-person noncooperative game in which each player’s strategy set depends on the rivals’ strategy set. In this paper, we presented a half-space projection method for solving the quasi-variational inequality problem which is a formulation of the GNEP. The difference from the known projection methods is due to the next iterate point in this method is obtained by directly projecting a point onto a half-space. Thus, our next iterate point can be represented explicitly. The global convergence is proved under the minimal assumptions. Compared with the known methods, this method can reduce one projection of a vector onto the strategy set per iteration. Numerical results show that this method not only outperforms the known method but is also less dependent on the initial value than the known method.     

Entropic perturbation method for solving a system of linear inequalities

The problem of finding an $\text{x∈}\text{R}{}^{\mathrm{n}}$ such that Ax⩽b and x⩾0 arises in numerous contexts. We propose a new optimization method for solving this feasibility problem. After converting Ax⩽b into a system of equations by introducing a slack variable for each of the linear inequalities, the method imposes an entropy function over both the original and the slack variables as the objective function. The resulting entropy optimization problem is convex and has an unconstrained convex dual. If the system is consistent and has an interior solution, then a closed-form formula converts the dual optimal solution to the primal optimal solution, which is a feasible solution for the original system of linear inequalities. An algorithm based on the Newton method is proposed for solving the unconstrained dual problem. The proposed algorithm enjoys the global convergence property with a quadratic rate of local convergence. However, if the system is inconsistent, the unconstrained dual is shown to be unbounded. Moreover, the same algorithm can detect possible inconsistency of the system. Our numerical examples reveal the insensitivity of the number of iterations to both the size of the problem and the distance between the initial solution and the feasible region. The performance of the proposed algorithm is compared to that of the surrogate constraint algorithm recently developed by Yang and Murty. Our comparison indicates that the proposed method is particularly suitable when the number of constraints is larger than that of the variables and the initial solution is not close to the feasible region.     

On the final steps of Newton and higher order methods

The Newton method is one of the most used methods for solving nonlinear system of equations when the Jacobian matrix is nonsingular. The method converges to a solution with Q-order two for initial points sufficiently close to the solution. The method of Halley and the method of Chebyshev are among methods that have local and cubic rate of convergence. Combining these methods with a backtracking and curvilinear strategy for unconstrained optimization problems these methods have been shown to be globally convergent. The backtracking forces a strict decrease of the function of the unconstrained optimization problem. It is shown that no damping of the step in the backtracking routine is needed close to a strict local minimizer and the global method behaves as a local method. The local behavior for the unconstrained optimization problem is investigated by considering problems with two unknowns and it is shown that there are no significant differences in the region where the global method turn into a local method for second and third order methods. Further, the final steps to reach a predefined tolerance are investigated. It is shown that the region where the higher order methods terminate in one or two iteration is significantly larger than the corresponding region for Newton’s method.     

An incomplete Hessian Newton minimization method and its application in a chemical database problem

To efficiently solve a large scale unconstrained minimization problem with a dense Hessian matrix, this paper proposes to use an incomplete Hessian matrix to define a new modified Newton method, called the incomplete Hessian Newton method (IHN). A theoretical analysis shows that IHN is convergent globally, and has a linear rate of convergence with a properly selected symmetric, positive definite incomplete Hessian matrix. It also shows that the Wolfe conditions hold in IHN with a line search step length of one. As an important application, an effective IHN and a modified IHN, called the truncated-IHN method (T-IHN), are constructed for solving a large scale chemical database optimal projection mapping problem. T-IHN is shown to work well even with indefinite incomplete Hessian matrices. Numerical results confirm the theoretical results of IHN, and demonstrate the promising potential of T-IHN as an efficient minimization algorithm.     

A proximal augmented Lagrangian method for equilibrium problems

Considering a recently proposed proximal point method for equilibrium problems, we construct an augmented Lagrangian method for solving the same problem in reflexive Banach spaces with cone constraints generating a strongly convergent sequence to a certain solution of the problem. This is an inexact hybrid method meaning that at a certain iterate, a solution of an unconstrained equilibrium problem is found, allowing a proper error bound, followed by a Bregman projection of the initial iterate onto the intersection of two appropriate halfspaces. Assuming a set of reasonable hypotheses, we provide a full convergence analysis.     

Convergence of the gradient projection method and Newton’s method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set

The gradient projection method and Newton’s method are generalized to the case of nonconvex constraint sets representing the set-theoretic intersection of a spherical surface with a convex closed set. Necessary extremum conditions are examined, and the convergence of the methods is analyzed.     

A proximal gradient descent method for the extended second-order cone linear complementarity problem

We consider an extended second-order cone linear complementarity problem (SOCLCP), including the generalized SOCLCP, the horizontal SOCLCP, the vertical SOCLCP, and the mixed SOCLCP as special cases. In this paper, we present some simple second-order cone constrained and unconstrained reformulation problems, and under mild conditions prove the equivalence between the stationary points of these optimization problems and the solutions of the extended SOCLCP. Particularly, we develop a proximal gradient descent method for solving the second-order cone constrained problems. This method is very simple and at each iteration makes only one Euclidean projection onto second-order cones. We establish global convergence and, under a local Lipschitzian error bound assumption, linear rate of convergence. Numerical comparisons are made with the limited-memory BFGS method for the unconstrained reformulations, which verify the effectiveness of the proposed method.     

Parallel implementation of Newton’s method for solving large-scale linear programs

Parallel versions of a method based on reducing a linear program (LP) to an unconstrained maximization of a concave differentiable piecewise quadratic function are proposed. The maximization problem is solved using the generalized Newton method. The parallel method is implemented in C using the MPI library for interprocessor data exchange. Computations were performed on the parallel cluster MVC-6000IM. Large-scale LPs with several millions of variables and several hundreds of thousands of constraints were solved. Results of uniprocessor and multiprocessor computations are presented.     

A smoothing homotopy method for variational inequality problems on polyhedral convex sets

In this paper, based on the Robinson’s normal equation and the smoothing projection operator, a smoothing homotopy method is presented for solving variational inequality problems on polyhedral convex sets. We construct a new smoothing projection operator onto the polyhedral convex set, which is feasible, twice continuously differentiable, uniformly approximate to the projection operator, and satisfies a special approximation property. It is computed by solving nonlinear equations in a neighborhood of the nonsmooth points of the projection operator, and solving linear equations with only finite coefficient matrices for other points, which makes it very efficient. Under the assumption that the variational inequality problem has no solution at infinity, which is a weaker condition than several well-known ones, the existence and global convergence of a smooth homotopy path from almost any starting point in $R^n$ are proven. The global convergence condition of the proposed homotopy method is same with that of the homotopy method based on the equivalent KKT system, but the starting point of the proposed homotopy method is not necessarily an interior point, and the efficiency is more higher. Preliminary test results show that the proposed method is practicable, effective and robust.     

An entire space polynomial-time algorithm for linear programming

We propose an entire space polynomial-time algorithm for linear programming. First, we give a class of penalty functions on entire space for linear programming by which the dual of a linear program of standard form can be converted into an unconstrained optimization problem. The relevant properties on the unconstrained optimization problem such as the duality, the boundedness of the solution and the path-following lemma, etc, are proved. Second, a self-concordant function on entire space which can be used as penalty for linear programming is constructed. For this specific function, more results are obtained. In particular, we show that, by taking a parameter large enough, the optimal solution for the unconstrained optimization problem is located in the increasing interval of the self-concordant function, which ensures the feasibility of solutions. Then by means of the self-concordant penalty function on entire space, a path-following algorithm on entire space for linear programming is presented. The number of Newton steps of the algorithm is no more than $O(nL\log (nL/ {\varepsilon }))$ , and moreover, in short step, it is no more than $O(\sqrt{n}\log (nL/{\varepsilon }))$ .     

Fast projection onto the ordered weighted ℓ1 norm ball

In this paper,we provide a finitely terminated yet efficient approach to compute the Euclidean projection onto the ordered weighted?₁(OWL1)norm ball.In particular,an efficient semismooth Newton method is proposed for solving the dual of a reformulation of the original projection problem.Global and local quadratic convergence results,as well as the finite termination property,of the algorithm are proved.Numerical comparisons with the two best-known methods demonstrate the efficiency of our method.In addition,we derive the generalized Jacobian of the studied projector which,we believe,is crucial for the future designing of fast second-order nonsmooth methods for solving general OWL1 norm constrained problems.     

Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non‐linear minimization problem, usually with a Tikhonov‐like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal‐to‐noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non‐linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright © 2002 John Wiley & Sons, Ltd.