Robust-to-Dynamics Optimization

A robust-to-dynamics optimization (RDO) problem is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and a feasible set $\Omega\subseteq\mathbb{R}^n$), and (ii) a dynamical system (a map $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$). Its goal is to minimize $f$ over the set $\mathcal{S}\subseteq\Omega$ of initial conditions that forever remain in $\Omega$ under $g$. The focus of this paper is on the case where the mathematical program is a linear program and the dynamical system is either a known linear map, or an uncertain linear map that can change over time. In both cases, we study a converging sequence of polyhedral outer approximations and (lifted) spectrahedral inner approximations to $\mathcal{S}$. Our inner approximations are optimized with respect to the objective function $f$ and their semidefinite characterization---which has a semidefinite constraint of fixed size---is obtained by applying polar duality to convex sets that are invariant under (multiple) linear maps. We characterize three barriers that can stop convergence of the outer approximations from being finite. We prove that once these barriers are removed, our inner and outer approximating procedures find an optimal solution and a certificate of optimality for the RDO problem in a finite number of steps. Moreover, in the case where the dynamics are linear, we show that this phenomenon occurs in a number of steps that can be computed in time polynomial in the bit size of the input data. Our analysis also leads to a polynomial-time algorithm for RDO instances where the spectral radius of the linear map is bounded above by any constant less than one. Finally, in our concluding section, we propose a broader research agenda for studying optimization problems with dynamical systems constraints, of which RDO is a special case

Low-Rank Boolean Matrix Approximation by Integer Programming

Low-rank approximations of data matrices are an important dimensionality reduction tool in machine learning and regression analysis. We consider the case of categorical variables, where it can be formulated as the problem of finding low-rank approximations to Boolean matrices. In this paper we give what is to the best of our knowledge the first integer programming formulation that relies on only polynomially many variables and constraints, we discuss how to solve it computationally and report numerical tests on synthetic and real-world data

Urban Insecurity and Security Expenditures in Turkiye

The majority of the world\u27s population now lives in urban areas, making cities targets of warfare and areas of insecurity in recent years. This has led to a growing interest in urban security and a shift towards the militarization of traditional police forces. However, the relationship between urban security and securitization is not well understood in the literature, and research on the implications of public resource allocation is limited. This study aims to investigate the effectiveness of public security spending in reducing insecurity in urban areas in Turkey, against the backdrop of increased securitization in the 2000s. The study uses data from 81 provinces between 2010 and 2018 and constructs two alternative indicators of insecurity. The results show that an increase in security spending reduces provincial insecurity, while public-private wage inequality and out-migration from insecure provinces increase insecurity. The paper discusses the implications of these findings for democracy and human rights

Recovering Dantzig-Wolfe Bounds by Cutting Planes

Dantzig-Wolfe (DW) decomposition is a well-known technique in mixed-integer programming (MIP) for decomposing and convexifying constraints to obtain potentially strong dual bounds. We investigate cutting planes that can be derived using the DW decomposition algorithm and show that these cuts can provide the same dual bounds as DW decomposition. More precisely, we generate one cut for each DW block, and when combined with the constraints in the original formulation, these cuts imply the objective function cut one can simply write using the DW bound. This approach typically leads to a formulation with lower dual degeneracy that consequently has a better computational performance when solved by standard MIP solvers in the original space. We also discuss how to strengthen these cuts to improve the computational performance further. We test our approach on the Multiple Knapsack Assignment Problem and the Temporal Knapsack Problem, and show that the proposed cuts are helpful in accelerating the solution time without the need to implement branch and price

Binary matrix factorisation and completion via integer programming

Binary matrix factorisation is an essential tool for identifying discrete patterns in binary data. In this paper we consider the rank-k binary matrix factorisation problem (k-BMF) under Boolean arithmetic: we are given an n × m binary matrix X with possibly missing entries and need to find two binary matrices A and B of dimension n × k and k × m respectively, which minimise the distance between X and the Boolean product of A and B in the squared Frobenius distance. We present a compact and two exponential size integer programs (IPs) for k-BMF and show that the compact IP has a weak LP relaxation, while the exponential size IPs have a stronger equivalent LP relaxation. We introduce a new objective function, which differs from the traditional squared Frobenius objective in attributing a weight to zero entries of the input matrix that is proportional to the number of times the zero is erroneously covered in a rank-k factorisation. For one of the exponential size IPs we describe a computational approach based on column generation. Experimental results on synthetic and real word datasets suggest that our integer programming approach is competitive against available methods for k-BMF and provides accurate low-error factorisations