scholarly journals νZ - Maximal Satisfaction with Z3

10.29007/jmxj ◽  
2018 ◽  
Author(s):  
Nikolaj Bjorner ◽  
Anh-Dung Phan
Keyword(s):  
Theorem Proving ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Satisfiability Modulo Theories ◽  
Objective Functions ◽  
Smt Solver ◽  
Smt Solvers ◽  
Free Variables ◽  
Linear Optimization Problems ◽  
Pareto Fronts

Satisfiability Modulo Theories, SMT, solvers are used inmany applications. These applications benefit from thepower of tuned and scalable theorem proving technologiesfor supported logics and specialized theory solvers.SMT solvers are primarily used to determine whether formulas are satisfiable.Furthermore, when formulas are satisfiable, many applications need modelsthat assign values to free variables.Yet, in many cases arbitrary assignments are insufficient,and what is really needed is an <i>optimal</i> assignmentwith respect to objective functions. So far, users of Z3,an SMT solver from Microsoft Research, build custom loopsto achieve objective values. This is no longer necessarywith νZ (new-Z, or max-Z), an extension within Z3 that letsusers formulate objective functions directly with Z3. Under the hood there is aportfolio of approaches for solving linear optimization problems overSMT formulas, MaxSMT, and their combinations. Objective functions are combinedas either Pareto fronts, lexicographically, or each objective is optimized independently.

Motion planning in crowded planar environments

Robotica ◽  
1999 ◽  
Vol 17 (4) ◽  
pp. 365-371 ◽  
Author(s):  
Yoav Lasovsky ◽  
Leo Joskowicz
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Degrees Of Freedom ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Local Geometry ◽  
Objective Functions ◽  
Linear Optimization Problems ◽  
Geometrically Complex ◽  
Crowded Environments

We present a new algorithm for fine motion planning in geometrically complex situations. Geometrically complex situations have complex robot and environment geometry, crowded environments, narrow passages and tight fits. They require complex robot motions with coupled degrees of freedom. The algorithm constructs a path by incrementally building a graph of linearized convex configuration space cells and solving a series of linear optimization problems with varying objective functions. Its advantages are that it better exploits the local geometry of narrow passages in configuration space, and that its complexity does not significantly increase as the clearance of narrow passages decreases. We demonstrate the algorithm on examples which other planners could not solve.

scholarly journals AVATAR Modulo Theories

10.29007/k6tp ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Nikolaj Bjorner ◽  
Martin Suda ◽  
Andrei Voronkov
Keyword(s):  
Theorem Proving ◽  
Satisfiability Modulo Theories ◽  
Theorem Prover ◽  
Ground Theory ◽  
Smt Solver ◽  
First Order ◽  
Proof Search ◽  
Smt Solvers ◽  
Sat Solver ◽  
A New Technique

This paper introduces a new technique for reasoning with quantifiers and theories. Traditionally, first-order theorem provers (ATPs) are well suited to reasoning with first-order problems containing many quantifiers and satisfiability modulo theories (SMT) solvers are well suited to reasoning with first-order problems in ground theories such as arithmetic. A recent development in first-order theorem proving has been the AVATAR architecture which uses a SAT solver to guide proof search based on a propositional abstraction of the first-order clause space. The approach turns a single proof search into a sequence of proof searches on (much) smaller sub-problems. This work extends the AVATAR approach to use a SMT solver in place of the SAT solver, with the effect that the first-order solver only needs to consider ground-theory-consistent sub-problems. The new architecture has been implemented using the Vampire theorem prover and Z3 SMT solver. Our experimental results, and the results of recent competitions, show that such a combination can be highly effective.

Multirow Intersection Cuts Based on the Infinity Norm

2021 ◽  
Author(s):  
Álinson S. Xavier ◽  
Ricardo Fukasawa ◽  
Laurent Poirrier
Keyword(s):  
Optimization Problems ◽  
Valid Inequalities ◽  
Linear Optimization ◽  
Computational Cost ◽  
General Purpose ◽  
Mixed Integer ◽  
Infinity Norm ◽  
Intersection Cuts ◽  
Linear Optimization Problems ◽  
Mixed Integer Linear Optimization

When generating multirow intersection cuts for mixed-integer linear optimization problems, an important practical question is deciding which intersection cuts to use. Even when restricted to cuts that are facet defining for the corner relaxation, the number of potential candidates is still very large, especially for instances of large size. In this paper, we introduce a subset of intersection cuts based on the infinity norm that is very small, works for relaxations having arbitrary number of rows and, unlike many subclasses studied in the literature, takes into account the entire data from the simplex tableau. We describe an algorithm for generating these inequalities and run extensive computational experiments in order to evaluate their practical effectiveness in real-world instances. We conclude that this subset of inequalities yields, in terms of gap closure, around 50% of the benefits of using all valid inequalities for the corner relaxation simultaneously, but at a small fraction of the computational cost, and with a very small number of cuts. Summary of Contribution: Cutting planes are one of the most important techniques used by modern mixed-integer linear programming solvers when solving a variety of challenging operations research problems. The paper advances the state of the art on general-purpose multirow intersection cuts by proposing a practical and computationally friendly method to generate them.

scholarly journals A new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methods

2007 ◽  
Vol 49 (2) ◽  
pp. 259-270 ◽  
Author(s):  
Keyvan Aminis ◽  
Arash Haseli
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Point Of View ◽  
Theory And Practice ◽  
Theoretical Point ◽  
Barrier Functions ◽  
Iteration Bound ◽  
Linear Optimization Problems

AbstractInterior-Point Methods (IPMs) are not only very effective in practice for solving linear optimization problems but also have polynomial-time complexity. Despite the practical efficiency of large-update algorithms, from a theoretical point of view, these algorithms have a weaker iteration bound with respect to small-update algorithms. In fact, there is a significant gap between theory and practice for large-update algorithms. By introducing self-regular barrier functions, Peng, Roos and Terlaky improved this gap up to a factor of log n. However, checking these self-regular functions is not simple and proofs of theorems involving these functions are very complicated. Roos el al. by presenting a new class of barrier functions which are not necessarily self-regular, achieved very good results through some much simpler theorems. In this paper we introduce a new kernel function in this class which yields the best known complexity bound, both for large-update and small-update methods.

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