Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters

The strategy for obtaining global convergence is based on the trust region approach. The merit function is a type of augmented Lagrangian. A new updating scheme is introduced for the penalty parameter, by means of which monotone increase is not necessary. Global convergence results are proved and numerical experiments are presented. Received May 31, 1995 / Revised version received December 12, 1997 Published online October 21, 1998     

Primal and dual convergence of a proximal point exponential penalty method for linear programming

We consider the diagonal inexact proximal point iteration where f(x,r)=c ^T x+r∑exp[(A _i x-b _i )/r] is the exponential penalty approximation of the linear program min{c ^T x:Ax≤b}. We prove that under an appropriate choice of the sequences λ _k , ε _k and with some control on the residual ν ^k , for every r _k →0⁺ the sequence u ^k converges towards an optimal point u ^∞ of the linear program. We also study the convergence of the associated dual sequence μ _i ^k =exp[(A _i u ^k -b _i )/r _k ] towards a dual optimal solution. Received: May 2000 / Accepted: November 2001?Published online June 25, 2002     

On the rate of convergence of sequential quadratic programming with nondifferentiable exact penalty function in the presence of constraint degeneracy

We analyze the convergence of a sequential quadratic programming (SQP) method for nonlinear programming for the case in which the Jacobian of the active constraints is rank deficient at the solution and/or strict complementarity does not hold for some or any feasible Lagrange multipliers. We use a nondifferentiable exact penalty function, and we prove that the sequence generated by an SQP using a line search is locally R-linearly convergent if the matrix of the quadratic program is positive definite and constant over iterations, provided that the Mangasarian-Fromovitz constraint qualification and some second-order sufficiency conditions hold. Received: April 28, 1998 / Accepted: June 28, 2001?Published online April 12, 2002     

A combined penalty function and gradient projection method for nonlinear programming

A new programming algorithm for nonlinear constrained optimization problems is proposed. The method is based on the penalty function approach and thereby circumyents the necessity to maintain feasibility at each iteration, but it also behaves much like the gradient projection method. Although only first-order information is used, the algorithm converges asymptotically at a rate which is independent of the magnitude of the penalty term; hence, unlike the simple gradient method, the asymptotic rate of the proposed method is not affected by the ill-conditioning associated with the introduction of the penalty term. It is shown that the asymptotic rate of convergence of the proposed method is identical with that of the gradient projection method.Dedicated to Professor M. R. HestenesThis research was supported by the National Science Foundation, Grant No. GK-16125.     

An exact penalty function for semi-infinite programming

This paper introduces a global approach to the semi-infinite programming problem that is based upon a generalisation of the ℓ₁ exact penalty function. The advantages are that the ensuing penalty function is exact and the penalties include all violations. The merit function requires integrals for the penalties, which provides a consistent model for the algorithm. The discretization is a result of the approximate quadrature rather than an a priori aspect of the model. This research was partially supported by Natural Sciences and Engineering Research Council of Canada grants A-8639 and A-8442. This paper was typeset using software developed at Bell Laboratories and the University of California at Berkeley.     

An exact penalty function method with global convergence properties for nonlinear programming problems

In this paper a new continuously differentiable exact penalty function is introduced for the solution of nonlinear programming problems with compact feasible set. A distinguishing feature of the penalty function is that it is defined on a suitable bounded open set containing the feasible region and that it goes to infinity on the boundary of this set. This allows the construction of an implementable unconstrained minimization algorithm, whose global convergence towards Kuhn-Tucker points of the constrained problem can be established.     

Solving a class of semidefinite programs via nonlinear programming

In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of SDP relaxations of combinatorial optimization problems with binary variables as well as other important SDP problems. We also derive gradient formulas for the objective function of the resulting nonlinear optimization problem and show that both function and gradient evaluations have affordable complexities that effectively exploit the sparsity of the problem data. This transformation, together with the efficient gradient formulas, enables the solution of very large-scale SDP problems by gradient-based nonlinear optimization techniques. In particular, we propose a first-order log-barrier method designed for solving a class of large-scale linear SDP problems. This algorithm operates entirely within the space of the transformed problem while still maintaining close ties with both the primal and the dual of the original SDP problem. Global convergence of the algorithm is established under mild and reasonable assumptions. Received: January 5, 2000 / Accepted: October 2001?Published online February 14, 2002     

Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the bounds arising from using the optimal basis. We also present explicit bounds for SDP using the Monteiro-Zhang family of search directions and specialize them to the AHO, H..K..M, and NT directions. Received: December 1999 / Accepted: January 2001?Published online March 22, 2001     

Nonlinear programming without a penalty function or a filter

A new method is introduced for solving equality constrained nonlinear optimization problems. This method does not use a penalty function, nor a filter, and yet can be proved to be globally convergent to first-order stationary points. It uses different trust-regions to cope with the nonlinearities of the objective function and the constraints, and allows inexact SQP steps that do not lie exactly in the nullspace of the local Jacobian. Preliminary numerical experiments on CUTEr problems indicate that the method performs well.      

Interior-point methods for nonconvex nonlinear programming: orderings and higher-order methods

The paper extends prior work by the authors on loqo, an interior point algorithm for nonconvex nonlinear programming. The specific topics covered include primal versus dual orderings and higher order methods, which attempt to use each factorization of the Hessian matrix more than once to improve computational efficiency. Results show that unlike linear and convex quadratic programming, higher order corrections to the central trajectory are not useful for nonconvex nonlinear programming, but that a variant of Mehrotra’s predictor-corrector algorithm can definitely improve performance. Received: May 3, 1999 / Accepted: January 24, 2000?Published online March 15, 2000     

一种双层规划的光滑化目标罚函数算法

论文研究了一种双层规划的光滑化目标罚函数算法,在一些条件下,证明了光滑化罚优化问题等价于原双层规划问题,而且,当下层规划问题是凸规划问题时, 给出了一个求解算法和收敛性证明.     

An exact lower order penalty function and its smoothing in nonlinear programming

In this article, we consider a lower order penalty function and its ε-smoothing for an inequality constrained nonlinear programming problem. It is shown that any strict local minimum satisfying the second-order sufficiency condition for the original problem is a strict local minimum of the lower order penalty function with any positive penalty parameter. By using an ε-smoothing approximation to the lower order penalty function, we get a modified smooth global exact penalty function under mild assumptions.     

On the convexity of the multiplicative potential and penalty functions and related topics

It is well known that a function f of the real variable x is convex if and only if (x,y)→yf(y ^-1 x),y>0 is convex. This is used to derive a recursive proof of the convexity of the multiplicative potential function. In this paper, we obtain a conjugacy formula which gives rise, as a corollary, to a new rule for generating new convex functions from old ones. In particular, it allows to extend the aforementioned property to functions of the form (x,y)→g(y)f(g(y)^-1 x) and provides a new tool for the study of the multiplicative potential and penalty functions. Received: June 3, 1999 / Accepted: September 29, 2000?Published online January 17, 2001     

Duality and martingales: a stochastic programming perspective on contingent claims

The hedging of contingent claims in the discrete time, discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program. Application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics, namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale. The model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims. However, we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options. An extension of the model to incorporate pre-existing liabilities and endowments reveals the reasons why buyers and sellers trade in options. The model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets. Received: June 5, 2000 / Accepted: July 12, 2001?Published online December 6, 2001     

Robustness of posynomial geometric programming optima

Received April 2, 1998 / Revised version received July 8, 1998 Published online November 24, 1998     

A dual-relax penalty function approach for solving nonlinear bilevel programming with linear lower level problem

The penalty function method, presented many years ago, is an important numerical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty function approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach.     

A trust region method based on interior point techniques for nonlinear programming

An algorithm for minimizing a nonlinear function subject to nonlinear inequality constraints is described. It applies sequential quadratic programming techniques to a sequence of barrier problems, and uses trust regions to ensure the robustness of the iteration and to allow the direct use of second order derivatives. This framework permits primal and primal-dual steps, but the paper focuses on the primal version of the new algorithm. An analysis of the convergence properties of this method is presented. Received: May 1996 / Accepted: August 18, 2000?Published online October 18, 2000     

A branch and cut algorithm for nonconvex quadratically constrained quadratic programming

We present a branch and cut algorithm that yields in finite time, a globally ε-optimal solution (with respect to feasibility and optimality) of the nonconvex quadratically constrained quadratic programming problem. The idea is to estimate all quadratic terms by successive linearizations within a branching tree using Reformulation-Linearization Techniques (RLT). To do so, four classes of linearizations (cuts), depending on one to three parameters, are detailed. For each class, we show how to select the best member with respect to a precise criterion. The cuts introduced at any node of the tree are valid in the whole tree, and not only within the subtree rooted at that node. In order to enhance the computational speed, the structure created at any node of the tree is flexible enough to be used at other nodes. Computational results are reported that include standard test problems taken from the literature. Some of these problems are solved for the first time with a proof of global optimality. Received December 19, 1997 / Revised version received July 26, 1999?Published online November 9, 1999     

The many facets of linear programming

We examine the history of linear programming from computational, geometric, and complexity points of view, looking at simplex, ellipsoid, interior-point, and other methods. Received: June 22, 2000 / Accepted: April 4, 2001?Published online October 2, 2001     

不等式约束优化问题的低阶精确罚函数的光滑化算法

对不等式约束优化问题提出了一个低阶精确罚函数的光滑化算法. 首先给出了光滑罚问题、非光滑罚问题及原问题的目标函数值之间的误差估计,进而在弱的假
设之下证明了光滑罚问题的全局最优解是原问题的近似全局最优解. 最后给出了一个基于光滑罚函数的求解原问题的算法,证明了算法的收敛性,并给出数值算例说明算法的可行性.