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Fundamental Domains for Symmetric Optimization: Construction and Search
... By proceeding recursively on the subgroup G (α) , we generate a fundamental domain after at most n iterations. Danielson [9] (see also [8]) independently proposes the same method, and observes that it can be used to construct a fundamental domain with n · d inequalities for the symmetries of a polytope with d facets in R n . We say that a fundamental domain obtained via this method is a generalized Dirichlet domain (GDD), as it generalizes the classical construction by Dirichlet [12]. ...
... In particular we create subgroup consistent fundamental domains based on a sequence of nested stabilizers of the G-action on R n . This construction, which was independently discovered by Danielson [8,9], generalizes Dirichlet domains, and hence k-fundamental domains, as well as the Schreier-Sims fundamental domain, presented in Sect. 4.2.1. ...
Breaking symmetries is a popular way of speeding up the branch-and-bound method for symmetric integer programs. We study fundamental domains, which are minimal and closed symmetry breaking polyhedra. Our long-term goal is to understand the relationship between the complexity of such polyhedra and their symmetry breaking capability. Borrowing ideas from geometric group theory, we provide structural properties that relate the action of the group with the geometry of the facets of fundamental domains. Inspired by these insights, we provide a new generalized construction for fundamental domains, which we call generalized Dirichlet domain. Our construction is recursive and exploits the coset decomposition of the subgroups that fix given vectors in Rn. We use this construction to analyze a recently introduced set of symmetry breaking inequalities by Salvagnin (Symmetry Breaking Inequalities from the Schreier-Sims table. In: International Conference on the Integration of Constraint Programming, Artificial Intelligence, and Operations Research, pp. 521–529, 2018) and Liberti and Ostrowski (J Global Opt 60:183–194, 2014), called Schreier-Sims inequalities. In particular, this shows that every permutation group admits a fundamental domain with less than n facets. We also show that this bound is tight. Finally, we prove that the fundamental domain defined by the Schreier-Sims inequalities can contain an exponential number of isomorphic binary vectors for a given permutation group G. This provides evidence of the lack of symmetry breaking effectiveness of this fundamental domain. Conversely, a suitably constructed GDD for this G has linearly many inequalities and contains unique representatives for isomorphic binary vectors.
This paper is concerned with the application of model predictive control (MPC) to large-scale linear dynamical systems with linear inequality constraints. A decomposition is proposed of such problems into sets of independent MPCs of lower dimensions that preserves all information about the system, cost function, and constraints. Different from previous work, the constraints are incorporated in the decomposition procedure, which is attained by generalizing a previously developed technique to simultaneously block diagonalize a set of matrices. This approach is applied to practical examples involving large-scale systems with inequality constraints. It is shown that the computational complexity and the CPU time required to solve the transformed MPC problems are lower than those required by the solution of the original MPC problem.
Motivated by recent interest in group-symmetry in semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of matrices, or equivalently the irreducible decomposition of the generated matrix
-algebra. The method is composed of numerical-linear algebraic computations such as eigenvalue computation, and automatically makes full use of the underlying algebraic structure, which is often an outcome of physical or geometrical symmetry, sparsity, and structural or numerical degeneracy in the given matrices. The main issues of the proposed approach are presented in this paper under some assumptions, while the companion paper gives an algorithm with full generality. Numerical examples of truss and frame designs are also presented.
Semidefinite programming (SDP) is one of the most active areas in mathematical programming, due to varied applications and
the availability of interior point algorithms. In this paper we propose a new pre-processing technique for SDP instances that
exhibit algebraic symmetry. We present computational results to show that the solution times of certain SDP instances may
be greatly reduced via the new approach.
KeywordsSemidefinite programming–Algebraic symmetry–Pre-processing–Interior point methods
This paper studies algorithms for the solution of mixed symmetric linear complementarity problems. The goal is to compute
fast and approximate solutions of medium to large sized problems, such as those arising in computer game simulations and American
options pricing. The paper proposes an improvement of a method described by Kocvara and Zowe (Numer Math 68:95–106, 1994) that combines projected Gauss–Seidel iterations with subspace minimization steps. The proposed algorithm employs
a recursive subspace minimization designed to handle severely ill-conditioned problems. Numerical tests indicate that the
approach is more efficient than interior-point and gradient projection methods on some physical simulation problems that arise
in computer game scenarios.
We introduce orbital branching, an effective branching method for integer programs containing a great deal of symmetry. The method is based on computing
groups of variables that are equivalent with respect to the symmetry remaining in the problem after branching, including symmetry
that is not present at the root node. These groups of equivalent variables, called orbits, are used to create a valid partitioning
of the feasible region that significantly reduces the effects of symmetry while still allowing a flexible branching rule.
We also show how to exploit the symmetries present in the problem to fix variables throughout the branch-and-bound tree. Orbital
branching can easily be incorporated into standard integer programming software. Through an empirical study on a test suite
of symmetric integer programs, the question as to the most effective orbit on which to base the branching decision is investigated.
The resulting method is shown to be quite competitive with a similar method known as isomorphism pruning and significantly better than a state-of-the-art commercial solver on symmetric integer programs.
KeywordsInteger programming–Symmetry–Branch-and-bound algorithms
If a mathematical program has many symmetric optima, solving it via Branch-and-Bound techniques often yields search trees
of disproportionate sizes; thus, finding and exploiting symmetries is an important task. We propose a method for automatically
finding the formulation group of any given Mixed-Integer Nonlinear Program, and for reformulating the problem by means of
static symmetry breaking constraints. The reformulated problem—which is likely to have fewer symmetric optima—can then be
solved via standard Branch-and-Bound codes such as CPLEX (for linear programs) and Couenne (for nonlinear programs). Our computational results include formulation group tables for the MIPLib3, MIPLib2003, GlobalLib
and MINLPLib instance libraries and solution tables for some instances in the aforementioned libraries.
KeywordsGroup–Symmetry–Mixed integer nonlinear programming–Branch and bound
In this paper we give evidence suggesting that MAX- CUT is NP-hard to approximate to within a factor of GW+ , for all > 0, where GW denotes the approximation ra- tio achieved by the Goemans-Williamson algorithm (14), GW :878567. This result is conditional, relying on two conjectures: a) the Unique Games conjecture of Khot (24); and, b) a very believable conjecture we call the Majority Is Stablest conjecture. These results indicate that the geomet- ric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to ( + )-approximate MAX-2SAT, where :943943 is the minimum of (2 + 2 )=(3 cos( )) on ( 2 ; ). Mo- tivated by our proof techniques, we show that if the MAX- 2CSP and MAX-2SAT problems are slightly restricted — in a way that seems to retain all their hardness — then they have ( GW )- and ( )-approximation algorithms, re- spectively. Though we are unable to prove the Majority Is Stablest conjecture, we give some partial results and indicate pos- sible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to ( 3 4 + 1 2 + )-approximate ( :909155), assuming only the Unique Games conjecture. We also discuss MAX-2CSP problems over non-boolean do-
This paper deals with exploiting symmetry for solving linear and integer
programming problems. Basic properties of linear representations of finite
groups can be used to reduce symmetric linear programming to solving linear
programs of lower dimension. Combining this approach with knowledge of the
geometry of feasible integer solutions yields an algorithm for solving highly
symmetric integer linear programs which only takes time which is linear in the
number of constraints and quadratic in the dimension.
There has been substantial work studying consensus problems for which there is a single common final state, although there are many real-world complex networks for which the complete consensus may be undesirable. More recently, the concept of group consensus whereby subsets of nodes are chosen to reach a common final state distinct from others has been developed, but the methods tend to be independent of the underlying network topology. Here, an alternative type of group consensus is achieved for which nodes that are “symmetric” achieve a common final state. The dynamic behavior may be distinct between nodes that are not symmetric. We show how group consensus for heterogeneous linear agents can be achieved via a simple coupling protocol that exploits the topology of the network. We see that group consensus is possible on both stable and unstable trajectories. We observe and characterize the phenomenon of “isolated group consensus,” where one or more clusters may achieve group consensus while the other clusters do not.
This paper presents an alternating-direction method of multipliers (admm) algorithm for solving large-scale symmetric model predictive control (MPC) problems in real-time on embedded computers with limited computational and memory resources. Symmetry was used to find transformations of the states, inputs, and constraints of the mpc problem that decompose the dynamics and cost. We prove a key-property of the symmetric group that allows us to efficiently transform between the original and decomposed symmetric domains. This allows us to solve different sub-problems of a baseline admm algorithm in different domains where the computations are less expensive. This reduces the computational cost of each iteration from quadratic in problem size to linear. In addition, we show that our admm algorithm requires a constant amount of memory regardless of the problem size. We demonstrate our algorithm for a battery balancing problem which results in a reduction of computation-times from hours to seconds and a reduction in memory from hundreds of megabytes to tens of kilobytes.
The handling of symmetries in mixed integer programs in order to speed up the solution process of branch-and-cut solvers has recently received significant attention, both in theory and practice. This paper compares different methods for handling symmetries using a common implementation framework. We start by investigating the computation of symmetries and analyze the symmetries present in the MIPLIB 2010 instances. It turns out that many instances are affected by symmetry and most symmetry groups contain full symmetric groups as factors. We then present (variants of) six symmetry handling methods from the literature. Their implementation is tested on several testsets. On very symmetric instances used previously in the literature, it is essential to use methods like isomorphism pruning, orbital fixing, or orbital branching. Moreover, tests on the MIPLIB instances show that isomorphism pruning, orbital fixing, or adding symmetry breaking inequalities allow to speed-up the solution process by about 15% and more instances can be solved within the time limit.
This paper studies a class of problems consisting of minimizing a continuously differentiable function penalized with the so-called `0-norm over a symmetric set. These problems are hard to solve, yet prominent in many fields and applications. We first study the proximal mapping with respect to the `0-norm over symmetric sets, and provide an efficient method to attain it. The method is then improved for symmetric sets satisfying a sub-modularity-like property, which we call second order monotonicity (SOM). It is shown that many important symmetric sets, such as the `1, `2, `∞-balls, the simplex and the full-simplex, satisfy this SOM property. We then develop, under the validity of the SOM property, necessary optimality conditions, and corresponding algorithms that are guaranteed to converge to points satisfying the aforementioned optimality conditions. We prove the existence of a hierarchy between the optimality conditions, and consequently between the corresponding algorithms.
This paper extends previous results on symmetry in strictly convex linear model predictive control to non-strictly convex and nonlinear model predictive control. We define symmetry for constrained systems, controllers, and model predictive control problems. We show that symmetric model predictive control problems produce symmetric controllers. We show that the previously established methods of memory reduction can be applied to non-strictly convex problems. We apply these memory reduction techniques to the battery balancing problem. Exploiting symmetry leads to an exponential memory reduction and simple, intuitive optimal controllers.
We exploit symmetry to reduce the memory storage requirements of linear explicit model predictive controllers. In the first part of the paper we define controller symmetry. We describe how symmetry can be used to compress the explicit controller and discuss the implementation of the resulting compressed controller. In the second part we develop a method for computing the reduced-memory controller without first computing the full-memory controller by employing the concept of fundamental domain.
This paper studies the control of distributed storage networks with guarantees of constraints satisfaction and asymptotic stability. We consider two problems: network capacity maximization and network balancing. In the first part of the paper we describe the two problems, highlight their importance in a wide number of engineering applications, and compare them by analyzing the properties of their solutions. In the second part we present algorithms for solving both problems by using a convex one-step model predictive controller (MPC) which guarantees persistent state and flow constraints satisfaction. We present simple conditions which link the network topology, the MPC weights and the asymptotic stability of the closed-loop system. A numerical example illustrates the effectiveness of the proposed approach.
We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems. We make partic-ular emphasis on general duality properties as providing suboptimality or infeasibility certificates. Our focus is on the exciting developments occurred in the last few years, including robust optimization, combinatorial optimization, and algebraic methods such as sum-of-squares. These developments are illustrated with examples of applications to control systems.
The paper presents a branch-and-cut for solving (0, 1) integer linear programs having a large symmetry group. The group is
used for pruning the enumeration tree and for generating cuts. The cuts are non-standard, cutting integer feasible solutions
but leaving the optimal value of the problem unchanged. Pruning and cut generation are performed by backtracking procedures
using a Schreier-Sims table for representing the group. Applications to hard set covering problems and to the generation of
covering designs and error correcting codes are presented.
This paper describes components of a branch-and-cut algorithm for solving integer linear programs having a large symmetry group. It describes an isomorphism pruning algorithm and variable setting procedures using orbits of the symmetry group. Pruning and orbit computations are performed by backtracking procedures using a Schreier-Sims table for representing the symmetry group. Applications to hard set covering problems, generation of covering designs and error correcting codes are given.
Semidefinite programming (SDP) may be seen as a generalization of linear programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation.We survey three types of special structures in SDP data:1.A common ‘chordal’ sparsity pattern of all the data matrices. This structure arises in applications in graph theory, and may also be used to deal with more general sparsity patterns in a heuristic way.2.Low rank of all the data matrices. This structure is common in SDP relaxations of combinatorial optimization problems, and SDP approximations of polynomial optimization problems.3.The situation where the data matrices are invariant under the action of a permutation group, or, more generally, where the data matrices belong to a low dimensional matrix algebra. Such problems arise in truss topology optimization, particle physics, coding theory, computational geometry, and graph theory.We will give an overview of existing techniques to exploit these structures in the data. Most of the paper will be devoted to the third situation, since it has received the least attention in the literature so far.
In this paper we show how the symmetry present in many linear systems can be exploited to significantly reduce the computational effort required for controller synthesis. This approach may be applied when controller design specifications are expressible via semidefinite programming. In particular, when the overall system description is invariant under unitary coordinate transformations of the state space matrices, synthesis semidefinite programs can be decomposed into a collection of smaller semidefinite programs.
This paper explores the role of symmetries and reduction in nonlinear control
and optimal control systems. The focus of the paper is to give a geometric
framework of symmetry reduction of optimal control systems as well as to show
how to obtain explicit expressions of the reduced system by exploiting the
geometry. In particular, we show how to obtain a principal connection to be
used in the reduction for various choices of symmetry groups, as opposed to
assuming such a principal connection is given or choosing a particular symmetry
group to simplify the setting. Our result synthesizes some previous works on
symmetry reduction of nonlinear control and optimal control systems. Affine and
kinematic optimal control systems are of particular interest: We explicitly
work out the details for such systems and also show a few examples of symmetry
reduction of kinematic optimal control problems.
We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints, e.g., for a well-known formulation of the graph coloring problem.
We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining inequalities. For the cyclic group case, the descriptions turn out to be totally unimodular, while for the symmetric group case, both the description and the proof are more involved. The associated separation problems can be solved in linear time.
We show how to exploit symmetries of a graph to efficiently compute the fastest mixing Markov chain on the graph (i.e., find the transition probabilities on the edges to minimize the second-largest eigenvalue modulus of the transition probability matrix). Exploiting symmetry can lead to significant reduction in both the number of variables and the size of matrices in the corresponding semidefinite program, thus enable numerical solution of large-scale instances that are otherwise computationally infeasible. We obtain analytic or semi-analytic results for particular classes of graphs, such as edge-transitive and distance-transitive graphs. We describe two general approaches for symmetry exploitation, based on orbit theory and block-diagonalization, respectively. We also establish the connection between these two approaches. Comment: 39 pages, 15 figures