Asymptotic analysis of a path-following barrier method for linearly constrained convex problems *

In the present paper the logarithmic barrier method applied to the linearly constrained convex optimization problems is studied from the view point of classical path-following algorithms. In particular, the radius of convergence of Newton's method which depends on the barrier parameter itself is estimated in standard norms, being independent of the parameter, without explicitly using self-concordance properties. The obtained results establish a parameter selection rule which guarantees the overall convergence of a barrier technique with only one Newton step at each parameter level and the complexity of the method can be estimated.     

Insights into the interior-point methods

In this paper, we study the search directions of three important interior-point algorithms, namely, the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method. From an algebraic point of view, we show that the search directions of these three algorithms are merely Newton directions along three different paths that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming problem. From a geometric point of view, we show that these directions can be obtained by solving certain well-defined subproblems. Both views provide a general platform for studying the existing interior-point methods and deriving new interior-point algorithms. We illustrate the derivation of new interior-point algorithms by replacing the logarithmic barrier function with an entropic barrier function. The results have been generalized and discussed.This work is partially supported by the North Carolina Supercomputing Center 1990 Cray Grant Program sponsored by Cray Research.     

A Newton inexact interior-point method for large scale nonlinear optimization problems

In this paper, we describe a variant of the Newton Interior-Point method in [8] for nonlinear programming problems. In this scheme, the perturbation parameter can be chosen within a range of, values and we can use an iterative method for approximately solving the reduced linear system arising at each step. We have devised the inner termination rule which guarantees the global convergence of this Newton Inexact Interior-Point method. We remark that the required assumptions are weaker than those stated in [8], as shown by some numerical examples. This research was supported by the Italian Ministry for Education, University and Research (MIUR), FIRB Project No. RBAU01JYPN.     

Extensions of algorithmic senstivity calculations nonlinear programming using barrier and penalty functions

We provide a theoretical basis for approximating the sensitivity of a perturbed solution and the local optimalvalue function, using information generated by a sequential unconstrained minimization technique in the normal course of solving a mathematical program. We show that various algorithmic sensitivity results can be obtained without other assumptions than those needed for the corresponding nonalgorithmic results. Our results extend the algorithmic calculation of sensitivity information introduced by Fiacco, utilizing the logarithmic barrier function and quadratic penalty function     

A globally convergent constrained quasi-Newton method with an augmented lagrangian type penalty function

The recently proposed quasi-Newton method for constrained optimization has very attractive local convergence properties. To force global convergnce of the method, a descent method which uses Zangwill's penalty function and an exact line search has been proposed by Han. In this paper a new method which adopts a differentiable penalty function and an approximate line is presented. The proposed penalty function has the form of the augmented Lagrangian function. An algorithm for updating parameters which appear in the penalty function is described. Global convergence of the given method is proved.     

Matrix augmentation and structure preservation in linearly constrained control problems

Matrix augmentation is used for the inversion of bases associated with large linearly constrained control problems. It is shown how an efficient data structure can be maintained by keeping all state variables in the basis, and then nullifying some of them explicitly by using additional constraints. The proposed methodology, together with a basis updating scheme based on augmentation, forms the skeleton for an in-core algorithm using either the revised simplex method or the generalized reduced gradient method.     

On penalty methods for minimax problems

Penalty algorithms for constrained minimax problems typically involve a sequence of unconstrained approximates which is pointwise monotone in each variable. This paper generalizes convergence results for a wider class of algorithms while imposing conditions which are close to being minimal.Sponsored in part by a grant from the Norwegian Council of Scientific and Technological Research.     

Secant algorithms with nonmonotone trust region that employs fletcher penalty function for constrained optimization

In this paper, we present a family of secant algorithms in association with nonmonotone trust region strategy for nonlinear equality constrained optimization problems. The proposed algorithms are globally convergent while keeping the local superlinear rate by introducing Fletcher's penalty function as merit function.     

Projected gradient methods for linearly constrained problems

The aim of this paper is to study the convergence properties of the gradient projection method and to apply these results to algorithms for linearly constrained problems. The main convergence result is obtained by defining a projected gradient, and proving that the gradient projection method forces the sequence of projected gradients to zero. A consequence of this result is that if the gradient projection method converges to a nondegenerate point of a linearly constrained problem, then the active and binding constraints are identified in a finite number of iterations. As an application of our theory, we develop quadratic programming algorithms that iteratively explore a subspace defined by the active constraints. These algorithms are able to drop and add many constraints from the active set, and can either compute an accurate minimizer by a direct method, or an approximate minimizer by an iterative method of the conjugate gradient type. Thus, these algorithms are attractive for large scale problems. We show that it is possible to develop a finite terminating quadratic programming algorithm without non-degeneracy assumptions. Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38. Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38.     

Inverse optimization for linearly constrained convex separable programming problems

In this paper, we study inverse optimization for linearly constrained convex separable programming problems that have wide applications in industrial and managerial areas. For a given feasible point of a convex separable program, the inverse optimization is to determine whether the feasible point can be made optimal by adjusting the parameter values in the problem, and when the answer is positive, find the parameter values that have the smallest adjustments. A sufficient and necessary condition is given for a feasible point to be able to become optimal by adjusting parameter values. Inverse optimization formulations are presented with ℓ₁ and ℓ₂ norms. These inverse optimization problems are either linear programming when ℓ₁ norm is used in the formulation, or convex quadratic separable programming when ℓ₂ norm is used.     

A class of methods for linear programming

A class of methods is presented for solving standard linear programming problems. Like the simplex method, these methods move from one feasible solution to another at each iteration, improving the objective function as they go. Each such feasible solution is also associated with a basis. However, this feasible solution need not be an extreme point and the basic solution corresponding to the associated basis need not be feasible. Nevertheless, an optimal solution, if one exists, is found in a finite number of iterations (under nondegeneracy). An important example of a method in the class is the reduced gradient method with a slight modification regarding selection of the entering variable.     

Exact penalty property for a class of inequality-constrained minimization problems

We use the penalty approach in order to study constrained minimization problems. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish the exact penalty property for a large class of inequality-constrained minimization problems.     

An interior-proximal method for convex linearly constrained problems and its extension to variational inequalities

In this paper, an entropy-like proximal method for the minimization of a convex function subject to positivity constraints is extended to an interior algorithm in two directions. First, to general linearly constrained convex minimization problems and second, to variational inequalities on polyhedra. For linear programming, numerical results are presented and quadratic convergence is established.Corresponding author. His research has been supported by C.E.E grants: CI1* CT 92-0046.     

A path following algorithm for a class of convex programming problems

We present a primal-dual path following interior algorithm for a class of linearly constrained convex programming problems with non-negative decision variables. We introduce the definition of a Scaled Lipschitz Condition and show that if the objective function satisfies the Scaled Lipschitz Condition then, at each iteration, our algorithm reduces the duality gap by at least a factor of (1–/n), where is positive and depends on the curvature of the objective function, by means of solving a system of linear equations which requires no more than O(n³) arithmetic operations. The class of functions having the Scaled Lipschitz Condition includes linear, convex quadratic and entropy functions.     

On the classical logarithmic barrier function method for a class of smooth convex programming problems

In this paper, we describe a natural implementation of the classical logarithmic barrier function method for smooth convex programming. It is assumed that the objective and constraint functions fulfill the so-called relative Lipschitz condition, with Lipschitz constantM>0.In our method, we do line searches along the Newton direction with respect to the strictly convex logarithmic barrier function if we are far away from the central trajectory. If we are sufficiently close to this path, with respect to a certain metric, we reduce the barrier parameter. We prove that the number of iterations required by the algorithm to converge to an -optimal solution isO((1+M ²) log) orO((1+M ²)nlog), depending on the updating scheme for the lower bound.on leave from Eötvös University, Budapest, Hungary.     

A special newton-type optimization method

The Kuhn–Tucker conditions of an optimization problem with inequality constraints are transformed equivalently into a special nonlinear system of equations T ₀(z) = 0. It is shown that Newton's method for solving this system combines two valuable properties: The local Q-quadratic convergence without assuming the strict complementary slackness condition and the regularity of the Jacobian of T ₀ at a point z under reasonable conditions, so that Newton’s method can be used also far from a Kuhn–Tucker point     

Parametrizations of Kojima’s system and relations to penalty and barrier functions

We investigate two homotopies that perturb Kojima’s system for describing critical points of a nonlinear optimization problem in finite dimension. Each of them characterizes stationary points of a usual penalty and a new “barrier” function. The latter is a continuous deformation of the objective, symmetric to the penalty from a formal point of view. Stationary points of these functions appear as perturbed critical points and vice versa. This permits new interpretations of the related solution methods and allows estimates of the solutions by using implicit function theorems for Lipschitzian equations.     

Nonmonotone projected gradient methods based on barrier and Euclidean distances

We consider nonmonotone projected gradient methods based on non-Euclidean distances, which play the role of barrier for a given constraint set. Our basic scheme uses the resulting projection-like maps that produces interior trajectories, and combines it with the recent nonmonotone line search technique originally proposed for unconstrained problems by Zhang and Hager. The combination of these two ideas leads to produce a nonmonotone scheme for constrained nonconvex problems, which is proven to converge to a stationary point. Some variants of this algorithm that incorporate spectral steplength are also studied and compared with classical nonmonotone schemes based on the usual Euclidean projection. To validate our approach, we report on numerical results solving bound constrained problems from the CUTEr library collection.