Infeasibility analysis for systems of quadratic convex inequalities

In this paper we address the problem of the infeasibility of systems defined by quadratic convex inequality constraints. In particular, we investigate properties of irreducible infeasible sets and provide an algorithm that identifies a set of all constraints (K) that may affect the feasibility status of the system after some perturbation of the right-hand sides. We show that all irreducible sets, as well as infeasibility sets, are subsets of the set K, and that every infeasible system contains an inconsistent subsystem of cardinality not greater than the number of variables plus one. The results presented in this paper are also applicable to linear systems.     

Minimal infeasible constraint sets in convex integer programs

In this paper we investigate certain aspects of infeasibility in convex integer programs, where the constraint functions are defined either as a composition of a convex increasing function with a convex integer valued function of n variables or the sum of similar functions. In particular we are concerned with the problem of an upper bound for the minimal cardinality of the irreducible infeasible subset of constraints defining the model. We prove that for the considered class of functions, every infeasible system of inequality constraints in the convex integer program contains an inconsistent subsystem of cardinality not greater than 2 ⁿ , this way generalizing the well known theorem of Scarf and Bell for linear systems. The latter result allows us to demonstrate that if the considered convex integer problem is bounded below, then there exists a subset of at most 2 ⁿ −1 constraints in the system, such that the minimum of the objective function subject to the inequalities in the reduced subsystem, equals to the minimum of the objective function over the entire system of constraints.     

Analyzing infeasible nonlinear programs

Nonlinear optimizers often report infeasibility during the process of initial construction of a model, or alterations to an existing model. Because solvers are unable to decide feasibility of a nonlinear constraint set with perfect accuracy, there are numerous possible explanations: the physical model really is infeasible, there is an error in the nonlinear constraint set causing infeasibility, or the model is feasible but the initial point or solver parameters are poorly chosen. It is difficult to proceed to a diagnosis of the problem in a large NLP.This paper presents an algorithm providing automated assistance in analyzing infeasible NLPs. The deletion filtering algorithm isolates a Minimal Intractable Subsystem (MIS) of constraints, a minimal set of constraints which appears infeasible to the solver given a specified initial point and parameter settings. The MIS may be as small as a few constraints from among the very much larger set defining the original model, and helps to focus the examination, thereby speeding the diagnosis. A computer tool embodying the algorithm, LSGRG (MIS), is developed and applied to demonstration examples.     

A homogeneous smoothing-type algorithm for symmetric cone linear programs

In this paper, we investigate a smoothing-type algorithm for solving the symmetric cone linear program （（SCLP） for short） by making use of an augmented system of its optimality conditions. The algorithm only needs to solve one system of linear equations and to perform one line search at each iteration. It is proved that the algorithm is globally convergent without assuming any prior knowledge of feasibility/infeasibility of the problem. In particular, the algorithm may correctly detect solvability of （SCLP）. Furthermore, if （SCLP） has a solution, then the algorithm will generate a solution of （SCLP）, and if the problem is strongly infeasible, the algorithm will correctly detect infeasibility of （SCLP）.     

A new infeasible interior-point method based on Darvay’s technique for symmetric optimization

We present a full Nesterov and Todd step primal-dual infeasible interior-point algorithm for symmetric optimization based on Darvay’s technique by using Euclidean Jordan algebras. The search directions are obtained by an equivalent algebraic transformation of the centering equation. The algorithm decreases the duality gap and the feasibility residuals at the same rate. During this algorithm we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main iteration of the algorithm consists of a feasibility step and some centering steps. The starting point in the first iteration of the algorithm depends on a positive number ξ and it is strictly feasible for a perturbed pair. The feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close to the central path of the new perturbed pair. The algorithm finds an ?-optimal solution or detects infeasibility of the given problem. Moreover, we derive the currently best known iteration bound for infeasible interior-point methods.     

A repairing technique for the local search of the job-shop problem

The local search technique has become a widely used tool for solving many combinatorial optimization problems. In the case of the job-shop the implementation of such a technique is not straightforward at all due to the existence of the technological constraints among the operations that belong to the same job. Their presence renders a certain set of schedules infeasible. Consequently, special attention is required when defining optimization algorithms to prevent the possibility of reaching an infeasible schedule during execution. Traditionally, the problem is tackled on the neighborhood level by using only a limited set of moves for which feasibility inherently holds. This paper proposes an alternative way to avoid infeasibility by incorporating a repairing technique into the mechanism for applying moves to a schedule. Whenever an infeasible move is being applied, a repairing mechanism rearranges the underlying schedule in such a way that the feasibility of the move is restored. The possibility of reaching infeasible solutions is, therefore, eliminated on the lowest possible conceptual level. Consequently, neighborhood functions need not to be constrained to a limited set of feasible moves any more.     

多项式分裂可行问题

本文研究多项式分裂可行问题,即由多项式不等式定义的分裂可行问题,包括凸与非凸、可行与不可行的问题;给出多项式分裂可行问题解集的半定松弛表示;研究其半定松弛化问题的性质;并基于这些性质建立求解多项式分裂可行问题的半定松弛算法.本文在较为一般的条件下证明了,如果分裂可行问题有解,则可通过本文建立的算法求得一个解点;如果问题...     

On critical sets of a finite Moore family

Cluster collections obtained within the framework of most cluster structures studied in data analysis and classification are essentially Moore families. In this paper, we propose a simple intuitive necessary and sufficient condition for some subset of objects to be a critical set of a finite Moore family. This condition is based on a new characterization of quasi-closed sets. Moreover, we provide a necessary condition for a subset containing more than k objects (k ≥ 2) to be a critical set of a k-weakly hierarchical Moore family. Finally, as a consequence of this result, we identify critical sets of some k-weakly hierarchical Moore families and thereby generalize a result earlier obtained by Domenach and Leclerc in the particular case of weak hierarchies.     

Weak infeasibility in second order cone programming

The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a sequence of feasibility problems which mostly preserve the feasibility status of the original problem. This is used to show that for a given weakly infeasible problem at most m directions are needed to get arbitrarily close to the cone, where m is the number of Lorentz cones. We also tackle a closely related question and show that given a bounded optimization problem satisfying Slater’s condition, we may transform it into another problem that has the same optimal value but it is ensured to attain it. From solutions to the new problem, we discuss how to obtain solution to the original problem which are arbitrarily close to optimality. Finally, we discuss how to obtain finite certificate of weak infeasibility by combining our own techniques with facial reduction. The analysis is similar in spirit to previous work by the authors on SDPs, but a different approach is required to obtain tighter bounds on the number of directions needed to approach the cone.     

A note on “Efficient feasibility testing for dial-a-ride problems”

Hunsaker and Savelsbergh [B. Hunsaker, M.W.P. Savelsbergh, Efficient feasibility testing for dial-a-ride problems, Operations Research Letters 30 (2002) 169-173.] developed a linear-time algorithm to verify the feasibility for dial-a-ride problems. However, this algorithm may incorrectly declare infeasibility due to ride time constraints in some cases. We propose a revised procedure to address this flaw, but in an O(n²) worst-case time.     

Using group theory and transition matrices to study a class of metaheuristic neighborhoods

The one-step conjugative rearrangement neighborhood of all possible incumbent tours in an n-city single-agent Traveling Salesperson Problem is represented by a transition matrix. Using these matrices and employing group theory and the symmetric group on n letters, we show that all such matrices will fall into three different types: (1) irreducible matrices with one set of tours, (2) irreducible cyclic matrices of period 2 with two distinct sets of tours, and (3) reducible matrices with two equal-sized distinct sets of tours. In addition to giving the required conditions that yield each neighborhood type, we briefly discuss how these results are easily extended to multi-agent traveling salesperson problems and suggest directions for future investigations.     

Taketa’s theorem for some character degree sets

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then ${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality ${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if ${{\rm cd} {(G \mid N')}}$ is a set of non-trivial p–powers for some fixed prime p.     

An O(mn log (nU)) time algorithm to solve the feasibility problem

The phase I maximum flow and most positive cut methods are used to solve the feasibility problem. Both of these methods take one maximum flow computation. Thus the feasibility problem can be solved using maximum flow algorithms. Let n and m be the number of nodes and arcs, respectively. In this paper, we present an algorithm to solve the feasibility problem with integer lower and upper bounds. The running time of our algorithm is O(mn log (nU)), where U is the value of maximum upper bound. Our algorithm improves the O(m² log (nU))-time algorithm in [12]. Hence the current algorithm improves the running time in [12] by a factor of n. Sleator and Goldberg’s algorithm is one of the well-known maximum flow algorithms, which runs in O(mn log n) time, see [5]. Under similarity assumption [11], our algorithm runs in O(mn log n) time, which is the running time of Sleator and Goldberg’s algorithm. The merit of our algorithm is that, in the case of infeasibility of the given network, it not only diagnoses infeasibility but also presents some information that is useful to modeler in estimating the maximum cost of adjusting the infeasible network.     

A Full-Newton Step Infeasible Interior-Point Algorithm Based on Darvay Directions for Linear Optimization

We present a full-Newton step primal-dual infeasible interior-point algorithm based on Darvay’s search directions. These directions are obtained by an equivalent algebraic transformation of the centering equation. The algorithm decreases the duality gap and the feasibility residuals at the same rate. During this algorithm we construct strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. Each main iteration of the algorithm consists of a feasibility step and some centering steps. The starting point in the first iteration of the algorithm depends on a positive number ξ and it is strictly feasible for a perturbed pair, and feasibility steps find strictly feasible iterate for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterate close to the central path of the new perturbed pair. The algorithm finds an ?-optimal solution or detects infeasibility of the given problem. The iteration bound coincides with the best known iteration bound for linear optimization problems.     

Criteria for the stability of the finiteness property and for the uniqueness of Barabanov norms

A set of matrices is said to have the finiteness property if the maximal rate of exponential growth of long products of matrices drawn from that set is realised by a periodic product. The extent to which the finiteness property is prevalent among finite sets of matrices is the subject of ongoing research. In this article, we give a condition on a finite irreducible set of matrices which guarantees that the finiteness property holds not only for that set, but also for all sufficiently nearby sets of equal cardinality. We also prove a theorem giving conditions under which the Barabanov norm associated to a finite irreducible set of matrices is unique up to multiplication by a scalar, and show that in certain cases these conditions are also persistent under small perturbations.     

Robust least square semidefinite programming with applications

In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional second-order cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of second-order cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method (Birgin et al. in SIAM J. Optim. 10:1196–1211, 2000) to solve the dual problem. While it is well-known that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal infeasibility in situations when the primal feasible set lies in a known compact set. As an application, we consider robust correlation stress testing where data uncertainty arises due to untimely recording of portfolio holdings. In our computational experiments on this particular application, our algorithm performs reasonably well on medium-sized problems for real data when finding the optimal solution (if exists) or identifying primal infeasibility, and usually outperforms the standard interior-point solver SDPT3 in terms of CPU time.     

Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems

ln) iterations, where ν is the parameter of a self-concordant barrier for the cone, ε is a relative accuracy and ρ_f is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O(ln) iterations, where ρ_· is a primal or dual infeasibility measure. Received April 25, 1996 / Revised version received March 4, 1998 Published online October 9, 1998     

An infeasible-point subgradient method using adaptive approximate projections

We propose a new subgradient method for the minimization of nonsmooth convex functions over a convex set. To speed up computations we use adaptive approximate projections only requiring to move within a certain distance of the exact projections (which decreases in the course of the algorithm). In particular, the iterates in our method can be infeasible throughout the whole procedure. Nevertheless, we provide conditions which ensure convergence to an optimal feasible point under suitable assumptions. One convergence result deals with step size sequences that are fixed a priori. Two other results handle dynamic Polyak-type step sizes depending on a lower or upper estimate of the optimal objective function value, respectively. Additionally, we briefly sketch two applications: Optimization with convex chance constraints, and finding the minimum ? ₁-norm solution to an underdetermined linear system, an important problem in Compressed Sensing.     

Solution techniques for the Large Set Covering Problem

Given a finite set E and a family F={E₁,…,E_m} of subsets of E such that F covers E, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of F that also covers E. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that E and the subsets E_i are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of F that does not cover E. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the k-colouring problem which is formulated as a constraint satisfaction problem.     

Constrained derivative-free optimization on thin domains

Many derivative-free methods for constrained problems are not efficient for minimizing functions on “thin” domains. Other algorithms, like those based on Augmented Lagrangians, deal with thin constraints using penalty-like strategies. When the constraints are computationally inexpensive but highly nonlinear, these methods spend many potentially expensive objective function evaluations motivated by the difficulties in improving feasibility. An algorithm that handles this case efficiently is proposed in this paper. The main iteration is split into two steps: restoration and minimization. In the restoration step, the aim is to decrease infeasibility without evaluating the objective function. In the minimization step, the objective function f is minimized on a relaxed feasible set. A global minimization result will be proved and computational experiments showing the advantages of this approach will be presented.