A new condition number for linear programming

In this paper we define a new condition number ?(A) for the following problem: given a m by n matrix A, find x∈ℝ ⁿ , s.t. Ax<0. We characterize this condition number in terms of distance to ill-posedness and we compare it with existing condition numbers for the same problem. Received: November 5, 1999 / Accepted: November 2000?Published online September 17, 2001     

线性规划中的退化最优基图

Abstract. The basis graph G for a linear programming consists of all bases under pivot transfor-mations. A degenerate optimal basis graph G is a subgraph of G induced by all optimal bases ata degenerate optimal vertex x. In this paper, several conditions for the characterization of G“are presented.     

Exact solutions to linear programming problems

The use of floating-point calculations limits the accuracy of solutions obtained by standard LP software. We present a simplex-based algorithm that returns exact rational solutions, taking advantage of the speed of floating-point calculations and attempting to minimize the operations performed in rational arithmetic. Extensive computational results are presented.     

A characterization of Delsarte’s linear programming bound as a ratio bound

It is well known that the ratio bound is an upper bound on the stability number α(G) of a regular graph G. In this note it is proved that, if G is a graph whose edge is a union of classes of a symmetric association scheme, the Delsarte’s linear programming bound can alternatively be stated as the minimum of a set of ratio bounds. This result follows from a recently established relationship between a set of convex quadratic bounds on α(G) and the number ?′(G), a well known variant of the Lovász theta number, which was introduced independently by Schrijver [A. Schrijver, A comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory 25 (1979) 425-429] and McEliece et al. [R.J. McEliece, E.R. Rodemich, H.C. Rumsey Jr, The Lovász bound and some generalizations, J. Combin. Inform. System Sci. 3 (1978) 134-152].     

A sequential equality constrained quadratic programming algorithm for inequality constrained optimization

In this paper, the feasible type SQP method is improved. A new SQP algorithm is presented to solve the nonlinear inequality constrained optimization. As compared with the existing SQP methods, per single iteration, in order to obtain the search direction, it is only necessary to solve equality constrained quadratic programming subproblems and systems of linear equations. Under some suitable conditions, the global and superlinear convergence can be induced.     

A globally convergent interior point algorithm for non-convex nonlinear programming

In this paper, a new algorithm for tracing the combined homotopy path of the non-convex nonlinear programming problem is proposed. The algorithm is based on the techniques of β$\beta$-cone neighborhood and a combined homotopy interior point method. The residual control criteria, which ensures that the obtained iterative points are interior points, is given by the condition that ensures the β$\beta$-cone neighborhood to be included in the interior part of the feasible region. The global convergence and polynomial complexity are established under some hypotheses.     

A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints

This paper proposes a new algorithm for solving a type of complicated optimal power flow (OPF) problems in power systems, i.e., OPF problems with transient stability constraints (OTS). The OTS is converted into a semi-infinite programming (SIP) via some suitable function analysis. Then based on the KKT system of the reformulated SIP, a smoothing quasi-Newton algorithm is presented in which the numerical integration is used. The convergence of the algorithm is established. An OTS problem in power system is tested, which shows that the proposed algorithm is promising.     

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 $O(\sqrt{n}\log (nL/{\varepsilon }))$ .     

New trajectory-following polynomial-time algorithm for linear programming problems

A new interior point method for the solution of the linear programming problem is presented. It is shown that the method admits a polynomial time bound. The method is based on the use of the trajectory of the problem, which makes it conceptually very simple. It has the advantage above related methods that it requires no problem transformation (either affine or projective) and that the feasible region may be unbounded. More importantly, the method generates at each stage solutions of both the primal and the dual problem. This implies that, contrary to the simplex method, the quality of the present solution is known at each stage. The paper also contains a practical (i.e., deepstep) version of the algorithm.The author is indebted to J. Bisschop, P. C. J. M. Geven, J. H. Van Lint, J. Ponstein, and J. P. Vial for their remarks on an earlier version of this paper.     

Asymptotic analysis of the exponential penalty trajectory in linear programming

We consider the linear program min{cx: Axb} and the associated exponential penalty functionf _r(x) = cx + rexp[(A _ix – b_i)/r]. Forr close to 0, the unconstrained minimizerx(r) off _r admits an asymptotic expansion of the formx(r) = x ^* + rd^* + (r) wherex ^* is a particular optimal solution of the linear program and the error term(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectory(r) which converges exponentially fast to a particular dual optimal solution. These results are completed by an asymptotic analysis whenr tends to : the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution.Corresponding author. Both authors partially supported by FONDECYT.     

On two homogeneous self-dual approaches to linear programming and its extensions

We investigate the relation between interior-point algorithms applied to two homogeneous self-dual approaches to linear programming, one of which was proposed by Ye, Todd, and Mizuno and the other by Nesterov, Todd, and Ye. We obtain only a partial equivalence of path-following methods (the centering parameter for the first approach needs to be equal to zero or larger than one half), whereas complete equivalence of potential-reduction methods can be shown. The results extend to self-scaled conic programming and to semidefinite programming using the usual search directions. Received: July 1998 / Accepted: September 2000?Published online November 17, 2000     

A polynomial-time algorithm,based on Newton's method,for linear programming

A new interior method for linear programming is presented and a polynomial time bound for it is proven. The proof is substantially different from those given for the ellipsoid algorithm and for Karmarkar's algorithm. Also, the algorithm is conceptually simpler than either of those algorithms.This research was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship and by NSF Grant 8120790. The research was performed at the Mathematical Sciences Research Institute in Berkeley, California.     

Dual versus primal-dual interior-point methods for linear and conic programming

We observe a curious property of dual versus primal-dual path-following interior-point methods when applied to unbounded linear or conic programming problems in dual form. While primal-dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness, dual methods can sometimes implicitly move away from the analytic center of the set of infeasibility/unboundedness detectors. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

An infeasible-interior-point potential-reduction algorithm for linear programming

n . The method is based on Rockafellar’s proximal point algorithm and a cutting-plane technique. At each step, we use an approximate proximal point p^a(x_k) of x_k to define a v_k∈∂_εkf(p^a(x_k)) with ε_k≤α∥v_k∥, where α is a constant. The method monitors the reduction in the value of ∥v_k∥ to identify when a line search on f should be used. The quasi-Newton step is used to reduce the value of ∥v_k∥. Without the differentiability of f, the method converges globally and the rate of convergence is Q-linear. Superlinear convergence is also discussed to extend the characterization result of Dennis and Moré. Numerical results show the good performance of the method. Received October 3, 1995 / Revised version received August 20, 1998 Published online January 20, 1999     

Accelerating method of global optimization for signomial geometric programming

Signomial geometric programming (SGP) has been an interesting problem for many authors recently. Many methods have been provided for finding locally optimal solutions of SGP, but little progress has been made for global optimization of SGP. In this paper we propose a new accelerating method for global optimization algorithm of SGP using a suitable deleting technique. This technique offers a possibility to cut away a large part of the currently investigated region in which the globally optimal solution of SGP does not exist, and can be seen as an accelerating device for global optimization algorithm of SGP problem. Compared with the method of Shen and Zhang [Global optimization of signomial geometric programming using linear relaxation, Appl. Math. Comput. 150 (2004) 99–114], numerical results show that the computational efficiency is improved obviously by using this new technique in the number of iterations, the required saving list length and the execution time of the algorithm.     

Solving linear programs from sign patterns

This paper is an attempt to provide a connection between qualitative matrix theory and linear programming. A linear program is said to be sign-solvable if the set of sign patterns of the optimal solutions is uniquely determined by the sign patterns of A, b, and c. It turns out to be NP-hard to decide whether a given linear program is sign-solvable or not. We then introduce a class of sign-solvable linear programs in terms of totally sign-nonsingular matrices, which can be recognized in polynomial time. For a linear program in this class, we devise an efficient combinatorial algorithm to obtain the sign pattern of an optimal solution from the sign patterns of A, b, and c. The algorithm runs in O(mγ) time, where m is the number of rows of A and γ is the number of all nonzero entries in A, b, and c.     

Interior-point methods for linear optimization based on a kernel function with a trigonometric barrier term

In this paper, we present a new barrier function for primal–dual interior-point methods in linear optimization. The proposed kernel function has a trigonometric barrier term. It is shown that in the interior-point methods based on this function for large-update methods, the iteration bound is improved significantly. For small-update interior-point methods, the iteration bound is the best currently known bound for primal–dual interior-point methods.