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.     

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.     

A new polynomial-time algorithm for linear programming

We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n ^3.5 L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n ^2.5). We prove that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property. The ratio of the radius of the smallest sphere with center a′, containingP′ to the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time. This is a substantially revised version of the paper presented at the Symposium on Theory of Computing, Washington D. C., April 1984.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

Randomized pivot algorithms for P-matrix linear complementarity problems

We study orientations of the n-cube that come from simple principal pivot algorithms for the linear complementarity problem with a P-matrix. We show that these orientations properly generalize those that are obtained from linear objective functions on polytopes combinatorially equivalent to the cube. The orientations from LCP with a P-matrix may admit directed cycles. We give a sequence of problems on which the algorithm RANDOM-EDGE performs very badly. Received: February 12, 2001 / Accepted: September 9, 2001?Published online April 12, 2002     

Exact penalty functions and calmness for mathematical programming under nonlinear perturbations

In the present paper, the effects of nonlinear perturbations of constraint systems are considered over the relationship between calmness and exact penalization, within the context of mathematical programming with equilibrium constraints. Two counterexamples are provided showing that the crucial link between the existence of penalty functions and the property of calmness for perturbed problems is broken in the presence of general perturbations. Then, some properties from variational analysis are singled out, which are able to restore to a certain extent the broken link. Consequently, conditions on the value function associated to perturbed optimization problems are investigated in order to guarantee the occurrence of the above properties.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems

This paper is devoted to the study of nonsmooth generalized semi-infinite programming problems in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be locally Lipschitz. We introduce a constraint qualification which is based on the Mordukhovich subdifferential. Then, we derive a Fritz–John type necessary optimality condition. Finally, interrelations between the new and the existing constraint qualifications such as the Mangasarian–Fromovitz, linear independent, and the Slater are investigated.     

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].     

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 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.     

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     

Multistage bilevel programming problems

In this article, we first examine some modeling scenarios for a multistage bilevel programming problem and develop the solution techniques based on certain reformulations of the original problem. The optimality conditions obtained for a class of multistage problems are given in terms of the second order subdifferentials of Mordukhovich.     

On Hölder continuity of approximate solutions to parametric equilibrium problems

We establish verifiable sufficient conditions for Hölder continuity of approximate solutions to parametric equilibrium problems, when solutions may be not unique. Many examples are provided to illustrate the need of considering approximate solutions instead of exact solutions and the essentialness of the imposed assumptions. As applications, we derive this Hölder continuity for constrained minimization, variational inequalities and fixed point problems.     

Relations between stochastic and parametric programming for decision problems with a random objective function

In this paper we study the relation between the general concept for an optimal solution for stochastic programming problems with a random objective function-the concept of an £-efficient solution-and the associated parametric problem, We show that it is possible under certain assumptions to obtain some or even all £-efficient solutions of the stochastic problem by solving the parametric problem with respect to a certain parameter set.     

Partly convex programming and zermelo's navigation problems

Mathematical programs, that become convex programs after freezing some variables, are termed partly convex. For such programs we give saddle-point conditions that are both necessary and sufficient that a feasible point be globally optimal. The conditions require cooperation of the feasible point tested for optimality, an assumption implied by lower semicontinuity of the feasible set mapping. The characterizations are simplified if certain point-to-set mappings satisfy a sandwich condition.The tools of parametric optimization and basic point-to-set topology are used in formulating both optimality conditions and numerical methods. In particular, we solve a large class of Zermelo's navigation problems and establish global optimality of the numerical solutions.Research partly supported by NSERC of Canada.     

Tikhonov-type regularization method for efficient solutions in vector optimization

The paper is devoted to developing the Tikhonov-type regularization algorithm of finding efficient solutions to the vector optimization problem for a mapping between finite dimensional Hilbert spaces with respect to the partial order induced by a pointed closed convex cone. We prove that under some suitable conditions either the sequence generated by our method converges to an efficient solution or all of its cluster points belong to the set of all efficient solutions of this problem.     

The bounds of feasible space on constrained nonconvex quadratic programming

This paper presents a method to estimate the bounds of the radius of the feasible space for a class of constrained nonconvex quadratic programmings. Results show that one may compute a bound of the radius of the feasible space by a linear programming which is known to be a P$P$-problem [N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984) 373–395]. It is proposed that one applies this method for using the canonical dual transformation [D.Y. Gao, Canonical duality theory and solutions to constrained nonconvex quadratic programming, J. Global Optimization 29 (2004) 377–399] for solving a standard quadratic programming problem.     

Two modified HS type conjugate gradient methods for unconstrained optimization problems

Based on the modified secant equation, we propose two new HS type conjugate gradient formulas. Their forms are similar to the original HS conjugate gradient formula and inherit all nice properties of the HS method. By utilizing the technique of the three-term HS method in Zhang et al. (2007) [15], without the requirement of truncation and convexity of the objective function, we show that one with Wolfe line search and the other with Armijo line search are globally convergent. Moreover, under some mild conditions, the linear convergence rate of the two modified methods is established. The numerical results show that the proposed methods are efficient.     

A method combining norm-relaxed QP subproblems with systems of linear equations for constrained optimization

Based on the ideas of norm-relaxed sequential quadratic programming (SQP) method and the strongly sub-feasible direction method, we propose a new SQP algorithm for the solution of nonlinear inequality constrained optimization. Unlike the previous work, at each iteration, the norm-relaxed quadratic programming subproblem (NRQPS) in our algorithm only consists of the constraints corresponding to an estimate of the active set, and the high-order correction direction (used to avoid the Maratos effect) is obtained by solving a system of linear equations (SLE) which also only consists of such a subset of constraints and gradients. Moreover, the line search technique can effectively combine the initialization process with the optimization process, and therefore (if the starting point is not feasible) the iteration points always get into the feasible set after a finite number of iterations. The global convergence is proved under the Mangasarian–Fromovitz constraint qualification (MFCQ), and the superlinear convergence is obtained without assuming the strict complementarity. Finally, the numerical experiments show that the proposed algorithm is effective and promising for the test problems.