Barrier Function Method Research Articles

Impproving the rate of convergence of the logarithmic barrier function method

Consider a convergent sequence {x k} generated by the application of the logarithmic barrier function method to a smooth convex programming problem. In this paper we show how to construct an auxiliary sequence {[xbar]k } that converges superlinearly faster than the original sequence {x k}. We then use the auxiliary sequence to develop a superlinearly convergent long step, predictor-corrector version of the barrier function method. Limited computational experience with the method of centres, a special version of the barrier method, on quadratically constrained programs suggests that our method deserves further study

A new linesearch method for quadratically constrained convex programming

A new method is proposed for the linesearch procedure in logarithmic barrier function and other interior point methods for convex quadratically constrained quadratic programming problems, which includes linear and quadratic programming as special cases. The method consists in transforming the derivative of the function obtained in this linesearch, so that Newton's method for finding its unique root will be globally convergent. It is equally useful in solving a problem in a matrix theory, namely the computation of the eigenvalues of a matrix, perturbed by the addition of a matrix of rank one. An improved Newton method is also presented together with numerical results.

A note on using alternative second-order models for the subproblems arising in barrier function methods for minimization

Inequality constrained minimization problems are often solved by considering a sequence of parameterized barrier functions. Each barrier function is approximately minimized and the relevant parameters subsequently adjusted. It is common for the estimated solution to one barrier function problem to be used as a starting estimate for the next. However, this has unfortunate repercussions for the standard Newton-like methods applied to the barrier subproblem. In this note, we consider a class of alternative Newton methods which attempt to avoid such difficulties. Such schemes have already proved of use in the Harwell Subroutine Library quadratic programming codes {\tt VE14} and {\tt VE19}.

Statistical Physics Algorithms That Converge

In recent years there has been significant interest in adapting techniques from statistical physics, in particular mean field theory, to provide deterministic heuristic algorithms for obtaining approximate solutions to optimization problems. Although these algorithms have been shown experimentally to be successful there has been little theoretical analysis of them. In this paper we demonstrate connections between mean field theory methods and other approaches, in particular, barrier function and interior point methods. As an explicit example, we summarize our work on the linear assignment problem. In this previous work we defined a number of algorithms, including deterministic annealing, for solving the assignment problem. We proved convergence, gave bounds on the convergence times, and showed relations to other optimization algorithms.

The convergence of a modified barrier method for convex programming

We show, using elementary considerations, that a modified barrier function method for the solution of convex programming problems converges for any fixed positive setting of the barrier parameter. With mild conditions on the primal and dual feasible regions, we show how to use the modified barrier function method to obtain primal and dual optimal solutions, even in the presence of degeneracy. We illustrate the argument for convergence in the case of linear programming, and then generalize it to the convex programming case.

Inverse design of super-elliptic cooling passages in coated turbine blade airfoils

A highly accurate and reliable algorithm capable of performing automatic inverse design of coolant flow passage numbers, shapes, sizes, and locations inside coated solid objects has been developed. The user has the freedom to specify arbitrary temperatures and heat fluxes at the points on the outer surface of the object, and either temperatures or heat fluxes on the surfaces of the yet unknown coolant flow passages. The number of passages required could be guessed and the algorithm will automatically eliminate the unnecessary passages. The method allows even inexperienced designers to achieve an optimal configuration of coolant passages in a single computer run while satisfying user-specified manufacturing constraints that were incorporated via a barrier function method. The optimization algorithms used in this inverse design code were based on gradient search and on a modified Newton search. A simple method for escaping from local minima has been implemented that involves switching between two different formulations of the objective function. The optimal value of the gradient search parameter was found using a simple method of fitting a highly accurate spline through a set of points in the cost function/search parameter plane, and seeking out the value that will generate minimal error.

On the convergence rate of regularization methods for ill-posed extremal problems

(1) J(u)→ inf, u ∈ U , (2) U = {u ∈ U0 : gi(u) ≤ 0 , i = 1, . . . ,m, gi(u) = 0, i = m+ 1, . . . , s} , where U0 is a given set and J(u), g1(u), . . . , gs(u) are finite functions defined on U0. It is known [7, 8, 10, 21], that the problem (1)–(2) is generally unstable under perturbation of the functions J(u), gi(u); therefore solving it we have to use some regularization technique. In the following we will use classical regularization methods such as the method of stabilization, the method of residuals and the method of quasisolutions. These methods will be based on the extension of the feasible set, or on the penalty and barrier function methods. One can find the application of these methods for the problem (1)–(2) in [7, 8, 10, 21, 9, 11, 2, 12, 22, 3, 13, 14, 24, 15] where the convergence rate for the objective function values has also been investigated. In this paper, which is a continuation of the papers [8, 21], we survey the results published in [13, 14, 24, 15, 16, 23, 26, 4, 5, 17] on the convergence rate of minimizing sequences constructed by the mentioned regularization methods.

A Long-step barrier method for convex quadratic programming

In this paper we propose a long-step logarithmic barrier function method for convex quadratic programming with linear equality constraints. After a reduction of the barrier parameter, a series of long steps along projected Newton directions are taken until the iterate is in the vicinity of the center associated with the current value of the barrier parameter. We prove that the total number of iterations isO(źnL) orO(nL), depending on how the barrier parameter is updated.

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 ful...

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.

Large Step Path-Following Methods for Linear Programming, Part I: Barrier Function Method

The algorithm proposed in this paper is the classical logarithmic barrier function method with Newton–Raphson steps for the internal minimization of the penalized function. Polynomial behavior is obtained by stopping each internal cycle when the iterates approach the central trajectory. Each master iteration updates the penalty parameter by a constant factor, and the overall complexity bound depends on this factor: $O(nL)$ Newton iterations for an arbitrary constant, and $O(\sqrt{n} L)$ iterations for a constant dependent on $\sqrt{n} $.

Convergence rate for regularized arrier function methods

In this paper the regularized variants of barrier function methods are discussed. These variants are related to the following regularization methods: Tichonov-function method, method of residuals and method of quasisolutions. .A convergence rate estimation is given for each method if both the original minimization problem and the connected problem of finding R-normal sohlion posses The feature of strong compatibility.

REMARKS ON POTENTIAL VERSUS BARRIER FUNCTION METHODS FOR LINEAR PROGRAMMING

We will point out some relations between potential and barrier function methods for linear programming. Then, based on the relations, we will show that the classical logarithmic barrier function method for linear programming can be adjusted so that it generates the optimal solution in O(√<n>L) iterations, where n is the number of variables and L is the data length. The method can be seen as a barrier function version of Ye's "An O(n^3L) potential reduction algorithm for linear programming".

An ?-active barrier-function method for solving minimax problems

A modification of an existing barrier-function method is presented. The modified algorithm may be used to solve semi-infinite minimax problems arising in engineering design. The modification preserves the global convergence properties, simple structure, and numerical robustness of the original algorithm, while substantially reducing the computational cost.

The modified barrier function method for linear programming and its extensions

We applied the modified barrier function (MBF) method and its new extensions to the linear programming (LP) problem and compared with the known LP barrier method. We then extended to the MBF LP method the worst case estimate O( L√ m) for the computational cost of the solution of an LP problem (having input size L and the number of constraints and variables m) in terms of the number of Newton's steps (such an estimate is well-known for the LP barrier methods), and we showed the improved numerical stability of Newton's steps compared with the LP barrier methods.

Local Convergence Properties of New Methods in Linear Programming

Asymptotic behavior of some of the interior point methods for Linear Programming is investigated without assuming nondegeneracy of constraints. A detailed analysis is given to the fundamental pair of vectors, the Newton direction leading to the center of the problem "d_C" and the direction of the affine-scaling method "d_<AF>". The quadratic convergence property of the iteration by - d_C is demonstrated. Step-sizes for the direction - d_C are also given to maintain feasibility without sacrificing the quadratic convergence. The sequence generated by - d_<AF> is pulled towards the central trajectory while. it converges linearly to the optimal solution. Local convergence properties of Iri and Imai's Multiplicative Barrier Function method and Yamashita's method (a variation of the projective scaling methods) are also discussed. It is shown that the search direction of the method by Iri and Imai converges to - d_C, whereas the direction by Yamashita converges to d_<AF>. A proof is given for the quadratic convergence property of Iri and Imai's method with an exact line search procedure in the case where constraint is degenerate.

Extended linear quadratic Gaussian control: Further extensions

The extended linear quadratic Gaussian (ELQG) control method (Georgakakos and Marks, 1987) is a stochastic control algorithm for the optimal operation of multiobjective reservoirs. Mathematically, this method optimizes a general functional of a stochastic system in state‐space form with upper and lower release constraints and probabilistic storage bounds. ELQG is a sequential algorithm which accounts for stochastic effects by preserving the first two statistical moments of the system's inputs and storages. In this paper, the method is first extended to handle nongaussian features which frequently characterize reservoir inputs. Second, ELQG's efficiency with respect to reliability storage constraints is discussed, and a new barrier function method is researched. These modifications are tested in case studies with the Savannah river system.

Algorithmic insights and a convergence analysis for a Karmarkar-type of algorithm for linear programming problems

This paper is concerned with a modification of a recently proposed variant of Karmarkar's algorithm for solving linear programming problems. In analyzing this variant, we exhibit interesting and useful relationships of these types of algorithms with barrier function methods, and subgradient optimization procedures involving space dilation techniques, which subsume the well-known ellipsoidal type of algorithms. Convergence of this variant is established under certain regularity conditions. We also provide remarks on how to obtain dual variables or Lagrange multipliers at optimality.

A multiplicative barrier function method for linear programming

A simple Newton-like descent algorithm for linear programming is proposed together with results of preliminary computational experiments on small- and medium-size problems. The proposed algorithm gives local superlinear convergence to the optimum and, experimentally, shows global linear convergence. It is similar to Karmarkar's algorithm in that it is an interior feasible direction method and self-correcting, while it is quite different from Karmarkar's in that it gives superlinear convergence and that no artificial extra constraint is introduced nor is protective geometry needed, but only affine geometry suffices.

Fitting three-parameter lognormal models by numerical global optimization — an improved algorithm

The theoretical maximum likelihood point estimates of the parameters of the three-parameter lognormal distribution (LD) for complete, ungrouped samples are infeasible values for which the likelihood function is positively infinite. This has prompted the search for usable point estimates which are called local-maximum likelihood estimates (LMLE's) and which correspond to the largest finite local maximum of the log-likelihood function (LLF). Previously published work has shown that an effective way to obtain LMLE's is to numerically maximize the conditional LLF of the observations using sequential interior penalty and barrier function methods. These methods are robust but necessitate the numerical solution of a sequence of maximization problems. This paper describes an algorithm for computing LMLE's which involves only a single numerical maximization of a suitably transformed conditional LLF. The results of computational tests indicate that the algorithm, when implemented using computationally efficient and reliable univariate global optimization methods, is a computationally robust method for computing LMLE's of LD parameters.