Infinite Linear Programming Research Articles

Continuous-Variable Nonlocality and Contextuality

Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete-variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine–Abramsky–Brandenburger theorem extends to this setting, an important consequence of which is that Bell nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, is also defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to discretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program. Through Lasserre relaxations, we are able to express this infinite linear program as a hierarchy of semi-definite programs that allow to calculate the contextual fraction with increasing accuracy.

A Simplex Method for Countably Infinite Linear Programs

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 2 December 2019Accepted: 26 July 2021Published online: 13 December 2021Keywordscountably infinite linear programs, infinite-dimensional optimization, simplex methodAMS Subject Headings90C49, 65K05Publication DataISSN (print): 1052-6234ISSN (online): 1095-7189Publisher: Society for Industrial and Applied MathematicsCODEN: sjope8

Matchmaking Strategies for Maximizing Player Engagement in Video Games

Managing player engagement is an important problem in the video game industry, as many games generate revenue via subscription models and microtransactions. We consider a class of online video games whereby players are repeatedly matched by the game to compete against one another. Players have different skill levels which affect the outcomes of matches, and the win-loss record influence their willingness to remain engaged. The goal is to maximize the overall player engagement over time by optimizing the dynamic matchmaking strategy. We propose a general but tractable framework to solve this problem, which can be formulated as an infinite linear program. We then focus on a stylized model where there are two skill levels and players churn only when they experience a losing streak. The optimal policy always matches as many low-skilled players who are not at risk of churning to high-skilled players who are one loss away from churning. In some scenarios when there are too many low-skilled players, high-skilled players are also matched to low-skilled players that are at risk of churning. Regarding the power of the optimal policy, we compare it to the industry status quo that matches players with the same skill level together (skill-based matchmaking). We prove the benefit of optimizing the matchmaking system grows linearly with the number of skill levels. We then use our framework to analyze two common but controversial interventions to increase engagement: adding AI bots and a pay-to-win system. We show that optimal matchmaking may reduce the number of bots needed significantly without loss of engagement. The pay-to-win system can influence player engagement positively when the majority of players are low-skilled. Surprisingly, even non-paying low-skilled players may be better off in some scenarios. Finally, we conduct a case study with real data from an online chess platform. We show the optimal policy can improve engagement by 4-6% or reducing the percentage of bot players by 15% in comparison to skill-based matchmaking.

Approximations of Countably Infinite Linear Programs over Bounded Measure Spaces

We study a class of countably infinite linear programs (CILPs) whose feasible sets are bounded subsets of appropriately defined spaces of measures. The optimal value, optimal points, and minimal po...

A note on primal-dual stability in infinite linear programming

In this note we analyze the simultaneous preservation of the consistency (and of the inconsistency) of linear programming problems posed in infinite dimensional Banach spaces, and their corresponding dual problems, under sufficiently small perturbations of the data. We consider seven different scenarios associated with the different possibilities of perturbations of the data (the objective functional, the constraint functionals, and the right hand-side function), i.e., which of them are known, and remain fixed, and which ones can be perturbed because of their uncertainty. The obtained results allow us to give sufficient and necessary conditions for the coincidence of the optimal values of both problems and for the stability of the duality gap under the same type of perturbations. There appear substantial differences with the finite dimensional case due to the distinct topological properties of cones in finite and infinite dimensional Banach spaces.

Inverse optimization in minimum cost flow problems on countably infinite networks

AbstractGiven the costs and a feasible solution for a minimum cost flow problem on a countably infinite network, inverse optimization involves finding new costs that are close to the original ones and that make the given solution optimal. We study this problem using the weighted absolute sum metric to quantify closeness of cost vectors. We provide sufficient conditions under which known results from inverse optimization in minimum cost flow problems on finite networks extend to the countably infinite case. These conditions ensure that recent duality results on countably infinite linear programs can be applied to our setting. Specifically, they enable us to prove that the inverse optimization problem can be reformulated as a capacitated, minimum cost circulation problem on a countably infinite network. Finite‐dimensional truncations of this problem can be solved in polynomial time when the weights equal one, which yields an efficient solution method. The circulation problem can also be solved via the shadow simplex method, where each finite‐dimensional truncation is tackled using the usual network Simplex algorithm that is empirically known to be computationally efficient. We illustrate these results on an infinite horizon shortest path problem.

On Infinite Linear Programming and the Moment Approach to Deterministic Infinite Horizon Discounted Optimal Control Problems

We revisit the linear programming approach to deterministic, continuous time, infinite horizon discounted optimal control problems. In the first part, we relax the original problem to an infinite-dimensional linear program over a measure space and prove equivalence of the two formulations under mild assumptions, significantly weaker than those found in the literature until now. The proof is based on duality theory and mollification techniques for constructing approximate smooth subsolutions to the associated Hamilton-Jacobi-Bellman equation. In the second part, we assume polynomial data and use Lasserre's hierarchy of primal-dual moment-sum-of-squares semidefinite relaxations to approximate the value function and design an approximate optimal feedback controller. We conclude with an illustrative example.

Infinite linear programming and online searching with turn cost

We consider the problem of searching for a hidden target in an environment that consists of a set of concurrent rays. Every time the searcher turns direction, it incurs a fixed cost. The objective is to derive a search strategy for locating the target as efficiently as possible, and the performance of the strategy is evaluated by means of the well-established competitive ratio. In this paper we revisit an approach due to Demaine et al. [8] based on infinite linear-programming formulations of this problem. We first demonstrate that their definition of duality in infinite LPs can lead to erroneous results. We then provide a non-trivial correction which establishes the optimality of a certain round-robin search strategy.

Optimal Signaling Mechanisms in Unobservable Queues

We consider the problem of optimal information sharing in an unobservable single-server queue offering service at a fixed price to a Poisson arrival of delay-sensitive customers. The service provider observes the queue, and may share state information with arriving customers. The customers are Bayesian and strategic, and incorporate this information into their beliefs before deciding whether to join the queue. We pose the following question: which signaling mechanism should the service provider adopt to maximize her expected revenue? We formulate this problem as an infinite linear program in the queue's steady-state distribution, and establish that, in general, the optimal signaling mechanism requires the service provider to strategically conceal information in order to incentivize customers to join. In particular, we show that a binary signaling mechanism with a threshold structure is optimal. Finally, we prove that coupled with an optimal fixed price, the optimal signaling mechanism generates the same expected revenue as the optimal state-dependent pricing mechanism. This suggests that in settings where state-dependent pricing is infeasible, signaling can be effective in achieving the optimal revenue. Our work contributes to the literature on dynamic Bayesian persuasion, and provides many interesting directions for extensions.

Duality in Countably Infinite Monotropic Programs

Finite-dimensional monotropic programs form a class of convex optimization problems that includes linear programs, convex minimum cost flow problems on networks and hypernetworks, and separable convex programs with linear constraints. Countably infinite monotropic programs arise, for example, in infinite-horizon sequential decision problems and in robust optimization. Their applications encompass (i) countably infinite linear programs such as the shortest path formulations of infinite-horizon nonstationary as well as countable-state Markov decision processes; and (ii) convex minimum cost flow problems on countably infinite networks and hypernetworks. Duality results for finite-dimensional monotropic programs are as powerful as those available for linear programs. On the contrary, applicable duality results for countably infinite monotropic programs are currently nonexistent owing to several mathematical pathologies in infinite-dimensional sequence spaces. This paper overcomes this hurdle by first embeddin...

Process intensification of reactive separator networks through the IDEAS conceptual framework

A method to rigorously identify the performance limits of a reactive separator network is presented in this paper. The quantification of the enhancement potential for a given technology can greatly benefit process intensification studies in the pursuit of radical improvements. The Infinite DimEnsionAl State-Space (IDEAS) conceptual framework is first reviewed and then shown to be capable of assessing the potential for process intensification of reactive separation processes. The IDEAS framework is employed to formulate an infinite linear program (ILP) that can synthesize optimal reactive separation networks, and establish rigorous tradeoffs between total network reactive holdup, and total network capacity. The proposed reactive separation process intensification method is demonstrated on a case study involving the metathesis of 2-pentene through reactive distillation. Significant intensification over prior designs is demonstrated.

Strategic health workforce planning

ABSTRACTAnalysts predict impending shortages in the health care workforce, yet wages for health care workers already account for over half of U.S. health expenditures. It is thus increasingly important to adequately plan to meet health workforce demand at reasonable cost. Using infinite linear programming methodology, we propose an infinite-horizon model for health workforce planning in a large health system for a single worker class; e.g., nurses. We give a series of common-sense conditions that any system of this kind should satisfy and use them to prove the optimality of a natural lookahead policy. We then use real-world data to examine how such policies perform in more complex systems; in particular, our experiments show that a natural extension of the lookahead policy performs well when incorporating stochastic demand growth.

On packing and covering polyhedra in infinite dimensions

We consider the natural generalizations of packing and covering polyhedra in infinite dimensions, and study issues related to duality and integrality of extreme points for these sets. Using appropriate finite truncations we give conditions under which complementary slackness holds for primal/dual pairs of the infinite linear programming problems associated with infinite packing and covering polyhedra. We also give conditions under which the extreme points are integral. We illustrate an application of our results on an infinite-horizon lot-sizing problem.

Robust optimization in countably infinite linear programs

A robust optimization framework for countably infinite linear programs (CILPs) is developed. It is shown that a particular robust counterpart of a nominal CILP can be reformulated as another CILP. A bound on the probability of constraint violation is derived. A convergent algorithm for solving this robust CILP is proposed.

Circumventing the Slater conundrum in countably infinite linear programs

Duality results on countably infinite linear programs are scarce. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix. Subspaces that permit a dual with this transpose do not admit an interior point. This difficulty has stumped researchers for a few decades; it has recently been called the Slater conundrum. We find a way around this hurdle.We propose a pair of primal-dual spaces with three properties: the series in the primal and dual objective functions converge; the series defined by the rows and columns of the primal constraint matrix converge; and the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose. Weak duality and complementary slackness are then immediate. Instead of using interior point conditions to establish a zero duality gap, we call upon the planning horizon method. When the series in the primal and dual constraints are continuous, we prove that strong duality holds if a sequence of optimal solutions to finite-dimensional truncations of the primal and dual CILPs has an accumulation point. We show by counterexample that the requirement that such an accumulation point exist cannot be relaxed. Our results are illustrated using several examples, and are applied to countable-state Markov decision processes and to a problem in robust optimization.

Inverse optimization in countably infinite linear programs

Given the costs and a feasible solution for a linear program, inverse optimization involves finding new costs that are close to the original ones and make the given solution optimal. We develop an inverse optimization framework for countably infinite linear programs using the weighted absolute sum metric. We reformulate this as an infinite-dimensional mathematical program using duality. We propose a convergent algorithm that solves a sequence of finite-dimensional LPs to tackle it. We apply this to non-stationary Markov decision processes.

Inverse Optimization in Countably Infinite Linear Programs

Given the objective coefficients and a feasible solution for a linear program, inverse optimization involves finding a new vector of objective coefficients that (i) is as close as possible to the original vector; and (ii) would make the given feasible solution optimal. This problem is well-studied for finite-dimensional linear programs. We develop a duality-based inverse optimization framework for countably infinite linear programs (CILPs) -- problems that include a countably infinite number of variables and constraints. Using the standard weighted absolute sum metric to quantify distance between cost vectors, we provide conditions under which constraints in the inverse optimization problem can be reformulated as a countably infinite set of linear constraints. We propose a convergent algorithm to solve the resulting infinite-dimensional mathematical program. This algorithm involves solving a sequence of finite-dimensional linear programs. We apply these results to inverse optimization in infinite-horizon non-stationary Markov decision processes.

Duality in Countably Infinite Linear Programs

Duality results on countably infinite linear programs (CILPs) are scarce. Roughly speaking, one fundamental hurdle is as follows. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual problem where the constraints cannot be expressed using the ordinary transpose of the primal doubly-infinite constraint matrix. Subspaces that do allow us to write the dual using this ordinary transpose do not admit an interior point. This has recently been called the "Slater conundrum." We find a way around this hurdle. We propose a pair of primal-dual spaces with three properties: (i) the series in the primal and dual objective functions converge; (ii) the series defined by the rows and columns of the primal constraint matrix converge; and (iii) the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose. Weak duality and complementary slackness are then immediate. Instead of using interior point conditions to establish a zero duality gap, we call upon the planning horizon method. When the series in the primal and dual constraints are continuous, we prove that strong duality holds if a sequence of optimal solutions to finite-dimensional truncations of the primal and dual CILPs has an accumulation point. We show by counterexample that the requirement that such an accumulation point exist cannot be relaxed. We also prove that the Lagrangean of our CILPs provides a linear support to the primal optimal value function. Finally, we establish that saddle points of the Lagrangean are equivalent to complementary feasible solutions of our CILPs. Our results are illustrated using several examples, and are also applied to countable-state Markov decision processes and to a problem in robust optimization.

Extreme point characterization of constrained nonstationary infinite-horizon Markov decision processes with finite state space

We study infinite-horizon nonstationary Markov decision processes with discounted cost criterion, finite state space, and side constraints. This problem can equivalently be formulated as a countably infinite linear program (CILP), a linear program with countably infinite number of variables and constraints. We provide a complete algebraic characterization of extreme points of the CILP formulation and illustrate the characterization for special cases. The existence of a K-randomized optimal policy for a problem with K side constraints also follows from this characterization.

On the attainable region for process networks

In this work the attainable region (AR) concept for process networks with outlet flow rate specifications is introduced for the first time. For process unit models to which the infinite dimensional State‐space conceptual framework is applicable, it is shown that identification of AR boundary membership is equivalent to feasibility assessment of an infinite linear program (ILP). A number of important AR properties are then theoretically established, including AR convexity, and representation of the AR in a concentration state space of reduced dimension. Finite dimensional approximations of the aforementioned ILP are then employed in creating increasingly accurate approximations of the AR. A case study for the vapor‐liquid equilibrium‐based separation of a ternary azeotropic mixture is used to illustrate the proposed method. The quantified two‐ and four‐Dimensional ARs indicate that acetone mole fractions above 0.79 (acetone/methanol azeotrope) are attainable for the considered outlet flow rate ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 193–212, 2014