Coordinate structures without syntactic categories

The issue of the syntactic category of unlike-category coordination has been elusive for decades, with a plethora of proposals, all deficient in one way or another. This chapter proposes to broaden the perspective and consider disjunctive constraints which are not limited to syntactic categories, but which also take into consideration morphosyntactic and lexical properties. Przepiórkowski and Patejuk present an account in which syntactic categories are encoded in functional-structures and all constraints on syntactic positions uniformly refer to functional-structures only. On this solution, the issue of syntactic categories of coordinate structures is void: same category coordinations have—via the definition of distributive properties—the same category as that of all the conjuncts, while unlike-category coordinations do not need—and, on this proposal, do not have—syntactic categories on top of the different categories of their conjuncts.

Ideal, non-extended formulations for disjunctive constraints admitting a network representation

In this paper we reconsider a known technique for constructing strong MIP formulations for disjunctive constraints of the form $$x \in \bigcup _{i=1}^m P_i$$ x ∈ ⋃ i = 1 m P i , where the $$P_i$$ P i are polytopes. The formulation is based on the Cayley Embedding of the union of polytopes, namely, $$Q := \mathrm {conv}(\bigcup _{i=1}^m P_i\times \{\epsilon ^i\})$$ Q : = conv ( ⋃ i = 1 m P i × { ϵ i } ) , where $$\epsilon ^i$$ ϵ i is the ith unit vector in $${\mathbb {R}}^m$$ R m . Our main contribution is a full characterization of the facets of Q, provided it has a certain network representation. In the second half of the paper, we work-out a number of applications from the literature, e.g., special ordered sets of type 2, logical constraints, the cardinality indicating polytope, union of simplicies, etc., along with a more complex recent example. Furthermore, we describe a new formulation for piecewise linear functions defined on a grid triangulation of a rectangular region $$D \subset {\mathbb {R}}^d$$ D ⊂ R d using a logarithmic number of auxilirary variables in the number of gridpoints in D for any fixed d. The series of applications demonstrates the richness of the class of disjunctive constraints for which our method can be applied.

Convex Relaxations for Quadratic On/Off Constraints and Applications to Optimal Transmission Switching

This paper studies mixed-integer nonlinear programs featuring disjunctive constraints and trigonometric functions and presents a strengthened version of the convex quadratic relaxation of the optimal transmission switching problem. We first characterize the convex hull of univariate quadratic on/off constraints in the space of original variables using perspective functions. We then introduce new tight quadratic relaxations for trigonometric functions featuring variables with asymmetrical bounds. These results are used to further tighten recent convex relaxations introduced for the optimal transmission switching problem in power systems. Using the proposed improvements, along with bound propagation, on 23 medium-sized test cases in the PGLib benchmark library with a relaxation gap of more than 1%, we reduce the gap to less than 1% on five instances. The tightened model has promising computational results when compared with state-of-the-art formulations.

Asymptotic stationarity and regularity for nonsmooth optimization problems

Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this new sense are already Mordukhovich-stationary, the presence of a constraint qualification which we call AM-regularity is necessary. We investigate the relationship between AM-regularity and other constraint qualifications from nonsmooth optimization like metric (sub-)regularity of the underlying feasibility mapping. Our findings are applied to optimization problems with geometric and, particularly, disjunctive constraints. This way, it is shown that AM-regularity recovers recently introduced cone-continuity-type constraint qualifications, sometimes referred to as AKKT-regularity, from standard nonlinear and complementarity-constrained optimization. Finally, we discuss some consequences of AM-regularity for the limiting variational calculus.

A Combinatorial Approach for Small and Strong Formulations of Disjunctive Constraints

A framework is presented for constructing strong mixed-integer programming formulations for logical disjunctive constraints. This approach is a generalization of the logarithmically sized formulations of Vielma and Nemhauser for special ordered sets of type 2 (SOS2) constraints, and a complete characterization of its expressive power is offered. The framework is applied to a variety of disjunctive constraints, producing novel small and strong formulations for outer approximations of multilinear terms, generalizations of special ordered sets, piecewise linear functions over a variety of domains, and obstacle avoidance constraints.

Replaceability for Constraint Satisfaction Problems: Algorithms, Inference, and Complexity Patterns

Replaceability is a form of generalized substitutability whose features make it potentially of great importance for problem simplification. It differs from simple substitutability in that it only requires that substitutable values exist for every solution containing a given value without requiring that the former always be the same. This is the most general form of substitutability that allows inferences from local to global versions of this property. Building on earlier work, this study first establishes that algorithms for localized replaceability (consistent neighbourhood replaceability or CNR algorithms) based on all-solutions neighbourhood search outperform other replaceability algorithms by several orders of magnitude. It also examines the relative effectiveness of different forms of depth-first CNR algorithms. Secondly, it demonstrates an apparent complexity ridge, which does not occur at the same place in the problem space as the complexity areas for consistency or full search algorithms. Thirdly, it continues the study of methods for inferring replaceability in structured problems in order to improve efficiency. Here, it is shown that some strategies for inferring replaceable values can be extended to disjunctive constraints in scheduling problems.

An iterative approach to precondition inference using constrained Horn clauses

We present a method for automatic inference of conditions on the initial states of a program that guarantee that the safety assertions in the program are not violated. Constrained Horn clauses (CHCs) are used to model the program and assertions in a uniform way, and we use standard abstract interpretations to derive an over-approximation of the set ofunsafeinitial states. The precondition then is the constraint corresponding to the complement of that set, under-approximating the set ofsafeinitial states. This idea of complementation is not new, but previous attempts to exploit it have suffered from the loss of precision. Here we develop an iterative specialisation algorithm to give more precise, and in some cases optimal safety conditions. The algorithm combines existing transformations, namely constraint specialisation, partial evaluation and a trace elimination transformation. The last two of these transformations perform polyvariant specialisation, leading to disjunctive constraints which improve precision. The algorithm is implemented and tested on a benchmark suite of programs from the literature in precondition inference and software verification competitions.