Arbitrary Linear Constraints Research Articles

Mixed integer linear programming formulation for K-means clustering problem

The minimum sum-of-squares clusering is the most widely used clustering method. The minimum sum-of-squares clustering is usually solved by the heuristic KMEANS algorithm, which converges to a local optimum. A lot of effort has been made to solve such kind of problems, but a mixed integer linear programming formulation (MILP) is still missing. In this paper, we formulate MILP models. The advantage of MILP formulation is that users can extend the original problem with arbitrary linear constraints. We also present numerical results, we solve these models up to sample size of 150.

On Lagrangian relaxation for constrained maximization and reoptimization problems

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small ε∈(0,1), we show that if there exists an r-approximation algorithm for the Lagrangian relaxation of the problem, for some r∈(0,1), then our technique achieves a ratio of rr+1−ε to the optimal, and this ratio is tight.Using the technique we obtain (re)approximation algorithms for natural (reoptimization) variants of classic subset selection problems, including real-time scheduling, the maximum generalized assignment problem (GAP) and maximum weight independent set. For all of the problems studied in this paper, the number of calls to the r-approximation algorithm, used by our algorithms, is linear in the input size and in log(1∕ε) for inputs with a cardinality constraint, and polynomial in the input size and in log(1∕ε) for inputs with an arbitrary linear constraint.

MIPaaL: Mixed Integer Program as a Layer

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures average accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a mixed integer linear program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, an algorithm that iteratively tightens the continuous relaxation by adding constraints removing fractional solutions. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and optimization separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP. Lastly, we demonstrate generalization performance in several transfer learning tasks.

Nested Convex Bodies are Chaseable

In the Convex Body Chasing problem, we are given an initial point v∈ Rd and an online sequence of n convex bodies F1, … , Fn. When we receive Ft, we are required to move inside Ft. Our goal is to minimize the total distance traveled. This fundamental online problem was first studied by Friedman and Linial (DCG 1993). They proved an Ω(d) lower bound on the competitive ratio, and conjectured that a competitive ratio depending only on d is possible. However, despite much interest in the problem, the conjecture remains wide open. We consider the setting in which the convex bodies are nested: F1⊃ ⋯ ⊃ Fn. The nested setting is closely related to extending the online LP framework of Buchbinder and Naor (ESA 2005) to arbitrary linear constraints. Moreover, this setting retains much of the difficulty of the general setting and captures an essential obstacle in resolving Friedman and Linial’s conjecture. In this work, we give a f(d)-competitive algorithm for chasing nested convex bodies in Rd.

Balancing and reconciling large multi-regional input\u2013output databases using parallel optimisation and high-performance computing

Over the past decade, large-scale multi-regional input–output (MRIO) tables have advanced the knowledge about pressing global issues. At the same time, the data reconciliation strategies required to construct such MRIOs have vastly increased in complexity: large-scale MRIOs are more detailed and hence require large amounts of different source data which are in return often varying in quality and reliability, overlapping, and, as a result, conflicting. Current MRIO reconciliation approaches—mainly RAS-type algorithms—cannot fully address this complexity adequately, since they are either tailored to handle certain classes of constraints only, or their mathematical foundations are currently unknown. Least-squares-type approaches have been identified to offer a robust mathematical framework, but the added complexity in terms of numerical handling and computing requirements has so far prevented the use of these methods for MRIO reconciliation tasks. We present a new algorithm (ACIM) based on a weighted least-squares approach. ACIM is able to reconcile MRIO databases of equal or greater size than then currently largest global MRIO databases. ACIM can address arbitrary linear constraints and consider lower and upper bounds as well as reliability information for each given data point. ACIM is based on judicious data preprocessing, state-of-the art quadratic optimisation, and high-performance computing in combination with parallel programming. ACIM addresses all shortcomings of RAS-type MRIO reconciliation approaches. ACIM’s was tested on the Eora model, and it was able to demonstrate improved runtimes and source data adherences.

SDP-based branch-and-bound for non-convex quadratic integer optimization

Semidefinite programming (SDP) relaxations have been intensively used for solving discrete quadratic optimization problems, in particular in the binary case. For the general non-convex integer case with box constraints, the branch-and-bound algorithm Q-MIST has been proposed by Buchheim and Wiegele (Math Program 141(1--2):435--452, 2013), which is based on an extension of the well-known SDP-relaxation for max-cut. For solving the resulting SDPs, Q-MIST uses an off-the-shelf interior point algorithm. In this paper, we present a tailored coordinate ascent algorithm for solving the dual problems of these SDPs. Building on related ideas of Dong (SIAM J Optim 26(3):1962--1985, 2016), it exploits the particular structure of the SDPs, most importantly a small rank of the constraint matrices. The latter allows both an exact line search and a fast incremental update of the inverse matrices involved, so that the entire algorithm can be implemented to run in quadratic time per iteration. Moreover, we describe how to extend this approach to a certain two-dimensional coordinate update. Finally, we explain how to include arbitrary linear constraints into this framework, and evaluate our algorithm experimentally.

Supervisory control of a class of Petri nets with unobservable and uncontrollable transitions

This paper uses Petri nets (PNs) as a modeling tool to deal with the forbidden state problem of discrete event systems (DESs) in the presence of both unobservable and uncontrollable events. First of all, it is proved that two state specifications are equivalent if their admissible marking sets coincide. Motivated by this result, we focus on studying how to compute optimal policies with respect to a state specification that is an admissible linear constraint. Thanks to many approaches in the literature that allow one to efficiently transform an arbitrary linear constraint into an admissible one with the admissible marking set unchanged, the proposed result remains useful in the more general case of arbitrary linear constraints. Specifically, focusing on ordinary PNs subject to an admissible linear constraint, we propose an optimal control policy whose computation mainly lies in the computation of the unobservable minimal decrease, a parameter depending on the current observation and the given constraint. A procedure to compute such a parameter with polynomial complexity is proposed provided that a particular subnet, called observation subnet, is acyclic and backward-conflict and backward-concurrent free (BBF). As a result, under such assumptions, the optimal control policy could be computed with polynomial complexity.

A mathematical framework for yield (vs. rate) optimization in constraint-based modeling and applications in metabolic engineering

Background: The optimization of metabolic rates (as linear objective functions) represents the methodical core of flux-balance analysis techniques which have become a standard tool for the study of genome-scale metabolic models. Besides (growth and synthesis) rates, metabolic yields are key parameters for the characterization of biochemical transformation processes, especially in the context of biotechnological applications. However, yields are ratios of rates, and hence the optimization of yields (as nonlinear objective functions) under arbitrary linear constraints is not possible with current flux-balance analysis techniques. Despite the fundamental importance of yields in constraint-based modeling, a comprehensive mathematical framework for yield optimization is still missing.Results: We present a mathematical theory that allows one to systematically compute and analyze yield-optimal solutions of metabolic models under arbitrary linear constraints. In particular, we formulate yield optimization as a linear-fractional program. For practical computations, we transform the linear-fractional yield optimization problem to a (higher-dimensional) linear problem. Its solutions determine the solutions of the original problem and can be used to predict yield-optimal flux distributions in genome-scale metabolic models. For the theoretical analysis, we consider the linear-fractional problem directly. Most importantly, we show that the yield-optimal solution set (like the rate-optimal solution set) is determined by (yield-optimal) elementary flux vectors of the underlying metabolic model. However, yield- and rate-optimal solutions may differ from each other, and hence optimal (biomass or product) yields are not necessarily obtained at solutions with optimal (growth or synthesis) rates. Moreover, we discuss phase planes/production envelopes and yield spaces, in particular, we prove that yield spaces are convex and provide algorithms for their computation. We illustrate our findings by a small example and demonstrate their relevance for metabolic engineering with realistic models of E. coli.Conclusions: We develop a comprehensive mathematical framework for yield optimization in metabolic models. Our theory is particularly useful for the study and rational modification of cell factories designed under given yield and/or rate requirements.

An algorithm based on semidefinite programming for finding minimax optimal designs

An algorithm based on a delayed constraint generation method for solving semi-infinite programs for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax optimization problem is solved using a semidefinite programming based approach that requires the design space be discretized. A nonlinear programming solver is then used to solve the inner program to determine the combination of the parameters that yields the worst-case value of the design criterion. The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it (i) produces designs that compare well with minimax D−optimal designs obtained from semi-infinite programming method in the literature; (ii) can be applied to semidefinite representable optimality criteria, that include the common A−,E−,G−,I− and D-optimality criteria; (iii) can tackle design problems with arbitrary linear constraints on the weights; and (iv) is fast and relatively easy to use.

From elementary flux modes to elementary flux vectors: Metabolic pathway analysis with arbitrary linear flux constraints.

Elementary flux modes (EFMs) emerged as a formal concept to describe metabolic pathways and have become an established tool for constraint-based modeling and metabolic network analysis. EFMs are characteristic (support-minimal) vectors of the flux cone that contains all feasible steady-state flux vectors of a given metabolic network. EFMs account for (homogeneous) linear constraints arising from reaction irreversibilities and the assumption of steady state; however, other (inhomogeneous) linear constraints, such as minimal and maximal reaction rates frequently used by other constraint-based techniques (such as flux balance analysis [FBA]), cannot be directly integrated. These additional constraints further restrict the space of feasible flux vectors and turn the flux cone into a general flux polyhedron in which the concept of EFMs is not directly applicable anymore. For this reason, there has been a conceptual gap between EFM-based (pathway) analysis methods and linear optimization (FBA) techniques, as they operate on different geometric objects. One approach to overcome these limitations was proposed ten years ago and is based on the concept of elementary flux vectors (EFVs). Only recently has the community started to recognize the potential of EFVs for metabolic network analysis. In fact, EFVs exactly represent the conceptual development required to generalize the idea of EFMs from flux cones to flux polyhedra. This work aims to present a concise theoretical and practical introduction to EFVs that is accessible to a broad audience. We highlight the close relationship between EFMs and EFVs and demonstrate that almost all applications of EFMs (in flux cones) are possible for EFVs (in flux polyhedra) as well. In fact, certain properties can only be studied with EFVs. Thus, we conclude that EFVs provide a powerful and unifying framework for constraint-based modeling of metabolic networks.

Stochastic disks that roll.

We study a model of rolling particles subject to stochastic fluctuations, which may be relevant in systems of nano- or microscale particles where rolling is an approximation for strong static friction. We consider the simplest possible nontrivial system: a linear polymer of three disks constrained to remain in contact and immersed in an equilibrium heat bath so the internal angle of the polymer changes due to stochastic fluctuations. We compare two cases: one where the disks can slide relative to each other and the other where they are constrained to roll, like gears. Starting from the Langevin equations with arbitrary linear velocity constraints, we use formal homogenization theory to derive the overdamped equations that describe the process in configuration space only. The resulting dynamics have the formal structure of a Brownian motion on a Riemannian or sub-Riemannian manifold, depending on if the velocity constraints are holonomic or nonholonomic. We use this to compute the trimer's equilibrium distribution with and without the rolling constraints. Surprisingly, the two distributions are different. We suggest two possible interpretations of this result: either (i) dry friction (or other dissipative, nonequilibrium forces) changes basic thermodynamic quantities like the free energy of a system, a statement that could be tested experimentally, or (ii) as a lesson in modeling rolling or friction more generally as a velocity constraint when stochastic fluctuations are present. In the latter case, we speculate there could be a "roughness" entropy whose inclusion as an effective force could compensate the constraint and preserve classical Boltzmann statistics. Regardless of the interpretation, our calculation shows the word "rolling" must be used with care when stochastic fluctuations are present.

Modeling Ion Channel Kinetics with Parameter Constraints

Ion channel gating mechanisms can be complex and difficult to extract from experimental data. A solution is to apply parameter constraints, which reflect prior knowledge or tested hypotheses and reduce model complexity and speed up computation. Soft constraints balance the existing knowledge with the new experimental data and limit the parameter search engine to a smaller space of more acceptable values. In contrast, hard constraints enforce a mathematical relationship involving one or more parameters of the model. These constraints can be formulated as an invertible transformation between a set of model parameters and a set of “free” parameters. Each constraint reduces the number of free parameters by one. Linear constraints, such as microscopic reversibility or scaling between sequential transitions, can be conveniently obtained with the singular value decomposition. Here, we show how this method can be generalized to implement arbitrary linear constraints. We also show how to make these constraints depend on arbitrary model parameters. This can be applied, for example, to enforce allosteric constraints where the allosteric factor itself is a free parameter. Furthermore, we explore some useful ways for implementing soft constraints.

Notes on ‘Hit-And-Run enables efficient weight generation for simulation-based multiple criteria decision analysis’

In our previous work published in this journal, we showed how the Hit-And-Run (HAR) procedure enables efficient sampling of criteria weights from a space formed by restricting a simplex with arbitrary linear inequality constraints. In this short communication, we note that the method for generating a basis of the sampling space can be generalized to also handle arbitrary linear equality constraints. This enables the application of HAR to sampling spaces that do not coincide with the simplex, thereby allowing the combined use of imprecise and precise preference statements. In addition, it has come to our attention that one of the methods we proposed for generating a starting point for the Markov chain was flawed. To correct this, we provide an alternative method that is guaranteed to produce a starting point that lies within the interior of the sampling space.

Bayesian area–age–period–cohort model with carcinogenesis age effects in estimating cancer mortality

Objective: Area–age–period–cohort (AAPC) model has been widely used in studying the spatial and temporal pattern of disease incidence and mortality rates. However, lack of biological plausibility and ease of interpretability on temporal components especially for age effects are generally the weakness of AAPC models. We develop a Bayesian AAPC model where carcinogenesis age effect is incorporated to explain age effects from the underlying disease process. An autoregressive prior structure and an arbitrary linear constraint are used to solve the nonidentifiability issues. Methods: Two multistage carcinogenesis models are employed to derive the hazard functions to substitute the age effects in the AAPC models. The Iowa county-wide lung cancer mortality data are used for the model fitting and Deviance Information Criteria (DIC) is used for model comparison. Results: Our study shows that conventional AAPC model (DIC=19,231.30), AAPC model with Armitage–Doll age effect (DIC=19,233.00) and with two-stage clonal expansion (TSCE) age effect (DIC=19,234.70) achieved the similar DIC values which indicated consistent model fitting among three models. The spatial pattern shows that the high spatial effects are clustered in the south of Iowa and also in largely populated areas. The lung cancer mortality rate is continuously declining by birth cohorts while increasing by the calendar period until 2000–2004. The age effects show an increasing pattern over time which can be easily explained by Armitage–Doll carcinogenesis model since we assume a log-linear relationship between age and hazard function. Conclusions: Our finding suggests that the proposed Bayesian AAPC model can be used to replace the conventional AAPC model without affecting model performance while providing a more biological sound approach from the underlining disease process.

Optimality Properties, Distributed Strategies, and Measurement-Based Evaluation of Coordinated Multicell OFDMA Transmission

The throughput of multicell systems is inherently limited by interference and the available communication resources. Coordinated resource allocation is the key to efficient performance, but the demand on backhaul signaling and computational resources grows rapidly with number of cells, terminals, and subcarriers. To handle this, we propose a novel multicell framework with dynamic cooperation clusters where each terminal is jointly served by a small set of base stations. Each base station coordinates interference to neighboring terminals only, thus limiting backhaul signalling and making the framework scalable. This framework can describe anything from interference channels to ideal joint multicell transmission. The resource allocation (i.e., precoding and scheduling) is formulated as an optimization problem (P1) with performance described by arbitrary monotonic functions of the signal-to-interference-and-noise ratios (SINRs) and arbitrary linear power constraints. Although (P1) is non-convex and difficult to solve optimally, we are able to prove: 1) Optimality of single-stream beamforming; 2) Conditions for full power usage; and 3) A precoding parametrization based on a few parameters between zero and one. These optimality properties are used to propose low-complexity strategies: both a centralized scheme and a distributed version that only requires local channel knowledge and processing. We evaluate the performance on measured multicell channels and observe that the proposed strategies achieve close-to-optimal performance among centralized and distributed solutions, respectively. In addition, we show that multicell interference coordination can give substantial improvements in sum performance, but that joint transmission is very sensitive to synchronization errors and that some terminals can experience performance degradations.

Characterization of SINR Region for Interfering Links With Constrained Power

In this paper, a communication system including n interfering additive white Gaussian noise (AWGN) links is considered. Each transmitter uses a Gaussian codebook and each receiver only decodes the data of the corresponding transmitter. For the case that the transmit powers are subject to arbitrary linear constraints, a mathematical expression for the boundary points of the signal-to-interference-plus-noise-ratio (SINR) region is obtained. Moreover, when the channels are time-varying and the average powers are constrained, the zero-outage SINR region of the system is derived. In addition, a scenario where the demanded SINR of the users is out of the SINR region is considered. A common approach is to remove a subset of the users such that the demanded SINR can be provided for the remaining users; the removed users are serviced in a later time slot. With the aim of maximizing the number of serviced users in each time slot, a suboptimal algorithm is developed, which outperforms the other known alternatives.

A least-constraint principle for population dynamics and reaction kinetics: Modeling entropy-controlled chemical hypercycles

In this paper, we investigate the treatment of constraints in rate equations describing the temporal evolution of biological populations or chemical reactions. We present a formulation for arbitrary holonomic and linear nonholonomic constraints which ensures the positivity of the dynamical variables and which is an analog of Gauss' principle of least constraint in classical mechanics. The approach is illustrated for the replication of molecular species in the Schuster-Eigen hypercycle model, imposing the conservation of the total number of molecules and the entropy production as constraints. The latter is used to model the behavior of an isolated system tending toward equilibrium and, for comparison, a stationary nonequilibrium state of an open system, which is characterized by undamped oscillations.

A kriging method for the solution of nonlinear programs with black‐box functions

AbstractIn this article, a new methodology is developed for the optimization of black‐box systems lacking a closed‐form mathematical description. To properly balance the computational cost of building the model against the probability of convergence to global optimum, a hybrid methodology is proposed. A kriging approach is first applied to provide information about the global behavior of the system considered, whereas a response surface method is considered close to the optimum to refine the set of candidate local optima and find the global optimum. The kriging predictor is a global model employing normally distributed basis functions, so both an expected sampling value and its variance are obtained for each test point. The presented work extends the capabilities of existing response surface techniques to address the refinement of optima located in regions described by convex asymmetrical feasible regions containing arbitrary linear and nonlinear constraints. The performance of the proposed algorithm is compared to previously developed stand‐alone response surface techniques and its effectiveness is evaluated in terms of the number of function calls required, number of times the global optimum is found, and computational time. © 2007 American Institute of Chemical Engineers AIChE J, 2007

P-SHAKE: A quadratically convergent SHAKE in [formula omitted

An algorithm for solving arbitrary linear constraints in molecular dynamics simulations of rigid and semi-rigid molecules is presented. The algorithm – P-SHAKE – is a modified version of the SHAKE [J.-P. Ryckaert, G. Ciccotti, H.J.C. Berendsen, Numerical integration of the cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes, J. Comput. Phys. 23 (1977) 327–341.] algorithm with a preconditioner applied which effectively de-couples the constraint equations. It achieves quadratic convergence, as does M-SHAKE [V. Kräutler, W.F. van Gunsteren, P.H. Hünenberger, A fast SHAKE algorithm to solve distance constraint equations for small molecules in molecular dynamics simulations. J. Comput. Chem. 22 (5) (2001) 501–508.], yet at a cost of only O ( n 2 ) operations per iteration, as opposed to O ( n 3 ) per iteration for M-SHAKE. The algorithm is applied to simulations of rigid water, DMSO, chlorophorm and non-rigid ethane and cyclohexane and is shown to be faster than M-SHAKE by up to a factor of three for relatively small error tolerances.

Uplink-downlink duality via minimax duality

The sum capacity of a Gaussian vector broadcast channel is the saddle point of a minimax Gaussian mutual information expression where the maximization is over the set of transmit covariance matrices subject to a power constraint and the minimization is over the set of noise covariance matrices subject to a diagonal constraint. This sum capacity result has been proved using two different methods, one based on decision-feedback equalization and the other based on a duality between uplink and downlink channels. This paper illustrates the connection between the two approaches by establishing that uplink-downlink duality is equivalent to Lagrangian duality in minimax optimization. This minimax Lagrangian duality relation allows the optimal transmit covariance and the least-favorable-noise covariance matrices in a Gaussian vector broadcast channel to be characterized in terms of the dual variables. In particular, it reveals that the least favorable noise is not unique. Further, the new Lagrangian interpretation of uplink-downlink duality allows the duality relation to be generalized to Gaussian vector broadcast channels with arbitrary linear constraints. However, duality depends critically on the linearity of input constraints. Duality breaks down when the input constraint is an arbitrary convex constraint. This shows that the minimax representation of the broadcast channel sum capacity is more general than the uplink-downlink duality representation.