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Jon Lee

Publications and source records attributed to Jon Lee.

At least 55 records · Page 3Linked to original sources
arXiv2023

On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose a branch-and-cut algorithm for the problem class we study, and a cutting-plane method for the problem variant with only binary variables. We present an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our binary instances.

math.OC↗
arXiv2022

Polynomial upper bounds on the number of differing columns of $Δ$-modular integer programs

We study integer-valued matrices with bounded determinants. Such matrices appear in the theory of integer programs (IP) with bounded determinants. For example, Artmann et al. showed that an IP can be solved in strongly polynomial time if the constraint matrix is bimodular, that is, the determinants are bounded in absolute value by two. Determinants are also used to bound the $\ell_1$-distance between IP solutions and solutions of its linear relaxation. One of the first works to quantify the complexity of IPs with bounded determinants was that of Heller, who identified the maximum number of differing columns in a totally unimodular matrix. Each extension of Heller's bound to general determinants has been super-polynomial in the determinants or the number of equations. We provide the first column bound that is polynomial in both values. For integer programs with box constraints, our result gives the first $\ell_1$-distance bound that is polynomial in the determinants and the number of equations. Our result can also be used to derive a bound on the height of Graver basis elements that is polynomial in the determinants and the number of equations. Furthermore, we show a tight bound on the number of differing columns in a bimodular matrix; this is the first tight bound since Heller. Our analysis reveals combinatorial properties of bimodular IPs that may be of independent interest.

math.OC↗
arXiv2022

D-optimal Data Fusion: Exact and Approximation Algorithms

We study the D-optimal Data Fusion (DDF) problem, which aims to select new data points, given an existing Fisher information matrix, so as to maximize the logarithm of the determinant of the overall Fisher information matrix. We show that the DDF problem is NP-hard and has no constant-factor polynomial-time approximation algorithm unless P $=$ NP. Therefore, to solve the DDF problem effectively, we propose two convex integer-programming formulations and investigate their corresponding complementary and Lagrangian-dual problems. We also develop scalable randomized-sampling and local-search algorithms with provable performance guarantees. Leveraging the concavity of the objective functions in the two proposed formulations, we design an exact algorithm, aimed at solving the DDF problem to optimality. We further derive a family of submodular valid inequalities and optimality cuts, which can significantly enhance the algorithm performance. Finally, we test our algorithms using real-world data on the new phasor-measurement-units placement problem for modern power grids, considering the existing conventional sensors. Our numerical study demonstrates the efficiency of our exact algorithm and the scalability and high-quality outputs of our approximation algorithms.

math.OC↗
arXiv2022

On computing with some convex relaxations for the maximum-entropy sampling problem

Based on a factorization of an input covariance matrix, we define a mild generalization of an upper bound of Nikolov (2015) and Li and Xie (2020) for the NP-Hard constrained maximum-entropy sampling problem (CMESP). We demonstrate that this factorization bound is invariant under scaling and also independent of the particular factorization chosen. We give a variable-fixing methodology that could be used in a branch-and-bound scheme based on the factorization bound for exact solution of CMESP. We report on successful experiments with a commercial nonlinear-programming solver. We further demonstrate that the known "mixing" technique can be successfully used to combine the factorization bound with the factorization bound of the complementary CMESP, and also with the "linx bound" of Anstreicher (2020).

math.OC↗
arXiv2022

SOCP-based disjunctive cuts for a class of integer nonlinear bilevel programs

We study a class of bilevel integer programs with second-order cone constraints at the upper level and a convex quadratic objective and linear constraints at the lower level. We develop disjunctive cuts to separate bilevel infeasible points using a second-order-cone-based cut-generating procedure. To the best of our knowledge, this is the first time disjunctive cuts are studied in the context of discrete bilevel optimization. Using these disjunctive cuts, we establish a branch-and-cut algorithm for the problem class we study, and a cutting plane method for the problem variant with only binary variables. We present a preliminary computational study on instances with no second-order cone constraints at the upper level and a single linear constraint at the lower level. Our study demonstrates that both our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our test instances, where the non-linearities are linearized in a McCormick fashion.

math.OC↗
arXiv2021

Experimental analysis of local searches for sparse reflexive generalized inverses

The well-known M-P (Moore-Penrose) pseudoinverse is used in several linear-algebra applications; for example, to compute least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given matrix is sparse, its M-P pseudoinverse can be completely dense, potentially leading to high computational burden and numerical difficulties, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but not all of them need to be satisfied for some applications. In this context, Fampa and Lee (Oper. Res. Letters, 46:605--610, 2018) and Xu, Fampa, Lee and Ponte (SIAM J. on Optimization, to appear) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy only some of the M-P properties. (Vector) 1-norm minimization is used to induce sparsity and to keep the magnitude of the entries under control, and theoretical results limit the distance between the 1-norm of the solution of the local searches and the minimum 1-norm of generalized inverses with corresponding properties. We have implemented several local-search procedures based on results presented in these two papers and make here an experimental analysis of them, considering their application to randomly generated matrices of varied dimensions, ranks, and densities. Further, we carried out a case study on a real-world data set.

math.OC↗
arXiv2021

Masking Anstreicher's linx bound for improved entropy bounds

The maximum-entropy sampling problem is the NP-hard problem of maximizing the (log) determinant of an order-$s$ principle submatrix of a given order $n$ covariance matrix $C$. Exact algorithms are based on a branch-and-bound framework. The problem has wide applicability in spatial statistics, and in particular in environmental monitoring. Probably the best upper bound for the maximum, empirically, is Anstreicher's scaled ``linx'' bound (see [K.M. Anstreicher. Efficient solution of maximum-entropy sampling problems. Oper. Res., 68(6):1826--1835, 2020]). An earlier methodology for potentially improving any upper-bounding method is by masking; i.e. applying the bounding method to $C\circ M$, where $M$ is any correlation matrix. We establish that the linx bound can be improved via masking by an amount that is at least linear in $n$, even when optimal scaling parameters are employed.

math.OC↗
arXiv2020

Trading off 1-norm and sparsity against rank for linear models using mathematical optimization: 1-norm minimizing partially reflexive ah-symmetric generalized inverses

The M-P (Moore-Penrose) pseudoinverse has as a key application the computation of least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given input matrix is sparse, its M-P pseudoinverse can be dense, potentially leading to high computational burden, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but only two of them need to be satisfied for the computation of least-squares solutions. Fampa and Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy the two key M-P properties, plus one more (the so-called reflexive property). That additional M-P property is equivalent to imposing a minimum-rank condition on the generalized inverse. (Vector) 1-norm minimization is used to induce sparsity and, importantly, to keep the magnitudes of entries under control for the generalized-inverses constructed. Here, we investigate the trade-off between low 1-norm and low rank for generalized inverses that can be used in the computation of least-squares solutions. We propose several algorithmic approaches that start from a $1$-norm minimizing generalized inverse that satisfies the two key M-P properties, and gradually decrease its rank, by iteratively imposing the reflexive property. The algorithms iterate until the generalized inverse has the least possible rank. During the iterations, we produce intermediate solutions, trading off low 1-norm (and typically high sparsity) against low rank.

math.OC↗
arXiv2020

Approximate 1-norm minimization and minimum-rank structured sparsity for various generalized inverses via local search

Fundamental in matrix algebra and its applications, a \emph{generalized inverse} of a real matrix $A$ is a matrix $H$ that satisfies the Moore-Penrose (M-P) property $AHA=A$. If $H$ also satisfies the additional useful M-P property, $HAH=H$, it is called a \emph{reflexive generalized inverse}. Reflexivity is equivalent to minimum rank, so we are particularly interested in reflexive generalized inverses. We consider aspects of symmetry related to the calculation of a \emph{sparse} reflexive generalized inverse of $A$. As is common, and following Lee and Fampa (2018) for calculating sparse generalized inverses, we use (vector) 1-norm minimization for inducing sparsity and for keeping the magnitude of entries under control. When $A$ is symmetric, we may naturally desire a symmetric $H$; while generally such a restriction on $H$ may not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically-efficient and practical local-search algorithm to block-construct an approximate 1-norm minimizing symmetric reflexive generalized inverse. Another aspect of symmetry that we consider relates to another M-P property: $H$ is \emph{ah-symmetric} if $AH$ is symmetric. The ah-symmetry property is the key one for solving least-squares problems using $H$. Here we do not assume that $A$ is symmetric, and we do not impose symmetry on $H$. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We provide a theoretically-efficient and practical local-search algorithm to column block construct an approximate 1-norm minimizing ah-symmetric reflexive generalized inverse.

math.OC↗
arXiv2020

1-norm minimization and minimum-rank structured sparsity for symmetric and ah-symmetric generalized inverses: rank one and two

Generalized inverses are important in statistics and other areas of applied matrix algebra. A \emph{generalized inverse} of a real matrix $A$ is a matrix $H$ that satisfies the Moore-Penrose (M-P) property $AHA=A$. If $H$ also satisfies the M-P property $HAH=H$, then it is called \emph{reflexive}. Reflexivity of a generalized inverse is equivalent to minimum rank, a highly desirable property. We consider aspects of symmetry related to the calculation of various \emph{sparse} reflexive generalized inverses of $A$. As is common, we use (vector) 1-norm minimization for both inducing sparsity and for keeping the magnitude of entries under control. When $A$ is symmetric, a symmetric $H$ is highly desirable, but generally such a restriction on $H$ will not lead to a 1-norm minimizing reflexive generalized inverse. We investigate a block construction method to produce a symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. Letting the rank of $A$ be $r$, we establish that the 1-norm minimizing generalized inverse of this type is a 1-norm minimizing symmetric generalized inverse when (i) $r=1$ and when (ii) $r=2$ and $A$ is nonnegative. Another aspect of symmetry that we consider relates to another M-P property: $H$ is \emph{ah-symmetric} if $AH$ is symmetric. The ah-symmetry property is sufficient for a generalized inverse to be used to solve the least-squares problem $\min\{\|Ax-b\|_2:~x\in\mathbb{R}^n\}$ using $H$, via $x:=Hb$. We investigate a column block construction method to produce an ah-symmetric reflexive generalized inverse that is structured and has guaranteed sparsity. We establish that the 1-norm minimizing ah-symmetric generalized inverse of this type is a 1-norm minimizing ah-symmetric generalized inverse when (i) $r=1$ and when (ii) $r=2$ and $A$ satisfies a technical condition.

math.OC↗
arXiv2020

An R Package for generating covariance matrices for maximum-entropy sampling from precipitation chemistry data

We present an open-source R package (MESgenCov v 0.1.0) for temporally fitting multivariate precipitation chemistry data and extracting a covariance matrix for use in the MESP (maximum-entropy sampling problem). We provide multiple functionalities for modeling and model assessment. The package is tightly coupled with NADP/NTN (National Atmospheric Deposition Program / National Trends Network) data from their set of 379 monitoring sites, 1978--present. The user specifies the sites, chemicals, and time period desired, fits an appropriate user-specified univariate model for each site and chemical selected, and the package produces a covariance matrix for use by MESP algorithms.

cs.MS↗
arXiv2020

Mixing convex-optimization bounds for maximum-entropy sampling

The maximum-entropy sampling problem is a fundamental and challenging combinatorial-optimization problem, with application in spatial statistics. It asks to find a maximum-determinant order-$s$ principal submatrix of an order-$n$ covariance matrix. Exact solution methods for this NP-hard problem are based on a branch-and-bound framework. Many of the known upper bounds for the optimal value are based on convex optimization. We present a methodology for "mixing" these bounds to achieve better bounds.

math.OC↗
arXiv2020

Improving proximity bounds using sparsity

We refer to the distance between optimal solutions of integer programs and their linear relaxations as proximity. In 2018, Eisenbrand and Weismantel proved that proximity is independent of the dimension for programs in standard form. We improve their bounds using existing and novel results on the sparsity of integer solutions. We first bound proximity in terms of the largest absolute value of any full-dimensional minor in the constraint matrix, and this bound is tight up to a polynomial factor in the number of constraints. We also give an improved bound in terms of the largest absolute entry in the constraint matrix, after efficiently transforming the program into an equivalent one. Our results are stated in terms of general sparsity bounds, so any new results on sparse solutions immediately improves our work. Generalizations to mixed integer programs are also discussed.

math.OC↗
arXiv2020

Gaining or losing perspective

We study MINLO (mixed-integer nonlinear optimization) formulations of the disjunction $x\in\{0\}\cup[l,u]$, where $z$ is a binary indicatorof $x\in[l,u]$ ($u> \ell > 0$), and $y$ "captures" $f(x)$, which is assumed to be convex on its domain $[l,u]$, but otherwise $y=0$ when $x=0$. This model is useful when activities have operating ranges, we pay a fixed cost for carrying out each activity, and costs on the levels of activities are convex. Using volume as a measure to compare convex bodies, we investigate a variety of continuous relaxations of this model, one of which is the convex-hull, achieved via the "perspective reformulation" inequality $y \geq zf(x/z)$. We compare this to various weaker relaxations, studying when they may be considered as viable alternatives. In the important special case when $f(x) := x^p$, for $p>1$, relaxations utilizing the inequality $yz^q \geq x^p$, for $q \in [0,p-1]$, are higher-dimensional power-cone representable, and hence tractable in theory. One well-known concrete application (with $f(x) := x^2$) is mean-variance optimization (in the style of Markowitz), and we carry out some experiments to illustrate our theory on this application.

math.OC↗
arXiv2018

Volume computation for sparse boolean quadric relaxations

Motivated by understanding the quality of tractable convex relaxations of intractable polytopes, Ko et al. gave a closed-form expression for the volume of a standard relaxation $\mathscr{Q}(G)$ of the boolean quadric polytope (also known as the (full) correlation polytope) $\mathscr{P}(G)$ of the complete graph $G=K_n$. We extend this work to structured sparse graphs, giving: (i) an efficient algorithm for $vol(\mathscr{Q}(G))$ when $G$ has bounded tree width, (ii) closed-form expressions (and asymptotic behaviors) for $vol(\mathscr{Q}(G))$ for all stars, paths, and cycles, and (iii) a closed-form expression for $vol(\mathscr{P}(G))$ for all cycles. Further, we demonstrate that when $G$ is a cycle, the simple relaxation $\mathscr{Q}(G)$ is a very close model for the much more complicated $\mathscr{P}(G)$. Additionally, we give some computational results demonstrating that this behavior of the cycle seems to extend to more complicated graphs. Finally, we speculate on the possibility of extending some of our results to cactii or even series-parallel graphs.

math.CO↗
arXiv2018

Approximating Max-Cut under Graph-MSO Constraints

We consider the max-cut and max-$k$-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a $\frac{1}{2}$-approximation algorithm for this class of problems.

cs.CC↗