Finite-dimensional Linear Programming Research Articles

Bi-attribute utility preference robust optimization: A continuous piecewise linear approximation approach

In this paper, we consider a bi-attribute decision making problem where the decision maker’s (DM’s) objective is to maximize the expected utility of outcomes with two attributes but where the true utility function which captures the DM’s risk preference is ambiguous. To tackle this ambiguity, we propose a maximin bi-attribute utility preference robust optimization (BUPRO) model where the optimal decision is based on the worst-case utility function in an ambiguity set of plausible utility functions constructed using partially available information such as the DM’s specific preference for certain lotteries. Specifically, we consider a BUPRO model with two attributes, where the DM’s risk attitude is bivariate risk-averse and the ambiguity set is defined by a linear system of inequalities represented by the Lebesgue–Stieltjes integrals of the DM’s utility functions. To solve the inner infinite-dimensional minimization problem, we propose a continuous piecewise linear approximation approach to approximate the DM’s unknown true utility. Unlike the univariate case, we partition the domain of the utility function into a set of small non-overlapping rectangles and then divide each rectangle into two triangles by either the main diagonal (Type-1) or the counter diagonal (Type-2). The inner minimization problem based on the piecewise linear utility function can be reformulated as a mixed-integer linear program and the outer maximization problem can be solved efficiently by the derivative-free method. In the case that all the small triangles are partitioned either in Type-1 or in Type-2, the inner minimization can be formulated as a finite dimensional linear program and the overall maximin as a single mixed-integer program. To quantify the approximation errors, we derive, under some mild conditions, the error bound for the difference between the BUPRO model and the approximate BUPRO model in terms of the ambiguity set, the optimal value and the optimal solutions. Finally, we carry out some numerical tests to examine the performance of the proposed models and computational schemes. The results demonstrate the efficiency of the computational schemes and highlight the stability of the BUPRO model against data perturbations.

Distributionally robust mean-absolute deviation portfolio optimization using wasserstein metric.

Data uncertainty has a great impact on portfolio selection. Based on the popular mean-absolute deviation (MAD) model, we investigate how to make robust portfolio decisions. In this paper, a novel Wasserstein metric-based data-driven distributionally robust mean-absolute deviation (DR-MAD) model is proposed. However, the proposed model is non-convex with an infinite-dimensional inner problem. To solve this model, we prove that it can be transformed into two simple finite-dimensional linear programs. Consequently, the problem can be solved as easily as solving the classic MAD model. Furthermore, the proposed DR-MAD model is compared with the 1/N, classic MAD and mean-variance model on S &P 500 constituent stocks in six different settings. The experimental results show that the portfolios constructed by DR-MAD model are superior to the benchmarks in terms of profitability and stability in most fluctuating markets. This result suggests that Wasserstein distributionally robust optimization framework is an effective approach to address data uncertainty in portfolio optimization.

Queueing Subject to Activity-Dependent Server Performance: Task-Assignment Control Policies for Utilization Rate Reduction

We consider a discrete-time system comprising a first-come-first-served queue, a nonpreemptive server, and a scheduler that assigns tasks from the queue to the server. New tasks enter the queue according to a Bernoulli process with a prespecified arrival rate. At each instant, the server is either working on a task or is available. The scheduler implements a task-assignment control policy that we seek to design. When the server is available and the queue is nonempty, the policy either assigns a new task to the server or allows it to remain available (to <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rest</i> ). In addition to the aforementioned <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">availability</i> state, we assume that the server has an integer-valued <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">activity state</i> . The activity state is nondecreasing during work periods, and is nonincreasing otherwise. In a typical application of our framework, the server performance (understood as task completion probability) worsens as the activity state increases. In this article, we build on and transcend recent stabilizability results obtained for the same framework. Specifically, we establish methods to design task-assignment control policies, which we call scheduler policies for short, which not only stabilize the queue but also reduce the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">utilization rate</i> —understood as the infinite-horizon time-averaged portion of time the server is working. This article has a main theorem leading to two key results: 1) We put forth a tractable method to determine, using a finite-dimensional linear program (LP), the infimum of all utilization rates that can be achieved by scheduler policies that are stabilizing, for a given arrival rate. 2) We propose a design method, also based on finite-dimensional LPs, to obtain stabilizing scheduler policies that can attain a utilization rate arbitrarily close to the aforementioned infimum. We also establish structural and distributional convergence properties, which are used throughout this article, and are significant in their own right.

Inverse conic linear programs in Banach spaces

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverse optimization involves finding new costs that are close to the original and that also render the given solution optimal. Ahuja and Orlin employed the absolute sum norm and the maximum absolute norm to quantify distances between cost vectors, and applied duality to establish that the inverse LP problem can be formulated as another finite-dimensional LP. This was recently extended to semi-infinite LPs, countably infinite LPs, and finite-dimensional conic optimization problems. These works provide sufficient conditions so that the inverse problem also belongs to the same class as the forward problem. This paper extends this result to conic LPs in potentially infinite-dimensional Banach spaces. Moreover, the paper presents concrete derivations for continuous conic LPs, whose special cases include continuous linear programs and continuous conic programs; normed cone programs in Banach spaces, which include second-order cone programs as a special case; and semi-definite programs in Hilbert spaces. These derivations reveal the sharper result that, in each case, the inverse problem belongs to the same specific subclass as the forward problem. Instances where existing forward algorithms can then be adapted to solve the inverse problems are identified. Results in this paper may enable the application of inverse optimization to as yet unexplored areas such as continuous-time economic planning, continuous-time job-shop scheduling, continuous-time network flow, maximum flow with time-varying edge-capacities, and wireless optimization with time-varying coverage requirements.

Nonlinear optimal control: a numerical scheme based on occupation measures and interval analysis

This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.

Inverse optimization in semi-infinite linear programs

Given the costs and a feasible solution for a finite-dimensional linear program (LP), inverse optimization involves finding new costs that are close to the original ones and render the given solution optimal. Ahuja and Orlin employed the absolute weighted sum metric to quantify distances between costs, and then applied duality to establish that inverse optimization reduces to another finite-dimensional LP. This paper extends this to semi-infinite linear programs (SILPs). A convergent Simplex algorithm to tackle the inverse SILP is proposed.

Robust continuous linear programs

Continuous linear programs (CLPs) arise in applications such as production/economic planning, continuous-time network flow problems, fluid relaxations of multiclass queueing networks, and control. To the best of the author’s knowledge, this paper proposes the first robust optimization framework for CLPs. The main result of the paper is that the robust counterpart of a CLP is also a CLP. Thus, any computational method for the original problem can be applied to the robust problem. For instance, a recent polynomial-time approximation algorithm applies. Further, if the original problem possesses a so-called separable structure, then the robust problem is also separable. Then existing Simplex-type and other discretization-based solution methods can be applied to the robust problem. The paper also provides a bound on the probability that an optimal solution to the robust counterpart violates a constraint in the original problem. Qualitative properties of this bound are discussed and compared with similar bounds for robust finite-dimensional linear programs.

Verification methods for conic linear programming problems

This paper gives an overview of verification methods for finite dimensional conic linear programming problems. Besides the computation of verified tight enclosures for unique non-degenerate solutions to well-posed conic linear programming instances, we discuss a rigorous treatment of problems with multiple or degenerate solutions. It will be further shown how a priori knowledge about certain boundedness qualifications can be exploited to efficiently compute verified bounds for the optimal objective value. The corresponding approach is applicable even to ill-posed programming problems. Examples from linear and semidefinite programming are used to illustrate the respective approaches and give further explanations. Another topic is the treatment of programming problems whose parameters are subject to uncertainties. Rough but inclusive estimates for the variability of the corresponding programming problems are given, and also a best and worst case analysis is taken into account. At the end of this paper, special consideration is given to typical issues when applying interval arithmetic in the context of conic linear programming. Different ways are shown on how to resolve these issues.

Computing Controlled Invariant Sets from Data Using Convex Optimization

This work presents a data-driven method for approximation of the maximum positively invariant (MPI) set and the maximum controlled invariant (MCI) set for nonlinear dynamical systems. The method only requires the knowledge of a finite collection of one-step transitions of the discrete-time dynamics, without the requirement of segments of trajectories or the control inputs that effected the transitions to be available. The approach uses a novel characterization of the MPI and MCI sets as the solution to an infinite-dimensional linear programming (LP) problem in the space of continuous functions, with the optimum being attained by a (Lipschitz) continuous function under mild assumptions. The infinite-dimensional LP is then approximated by restricting the decision variable to a finite-dimensional subspace and by imposing the non-negativity constraint of this LP only on the available data samples. This leads to a single finite-dimensional LP that can be easily solved using off-the-shelf solvers. We analyze the convergence rate and sample complexity, proving probabilistic as well as hard guarantees on the volume error of the approximations. The approach is very general, requiring minimal underlying assumptions. In particular, the dynamics is not required to be polynomial or even continuous (forgoing some of the theoretical results). Detailed numerical examples up to state-space dimension ten with code available online demonstrate the method.

On the forces that cable webs under tension can support and how to design cable webs to channel stresses.

In many applications of structural engineering, the following question arises: given a set of forces f 1, f 2, …, f N applied at prescribed points x 1, x 2, …, x N , under what constraints on the forces does there exist a truss structure (or wire web) with all elements under tension that supports these forces? Here we provide answer to such a question for any configuration of the terminal points x 1, x 2, …, x N in the two- and three-dimensional cases. Specifically, the existence of a web is guaranteed by a necessary and sufficient condition on the loading which corresponds to a finite dimensional linear programming problem. In two dimensions, we show that any such web can be replaced by one in which there are at most P elementary loops, where elementary means that the loop cannot be subdivided into subloops, and where P is the number of forces f 1, f 2, …, f N applied at points strictly within the convex hull of x 1, x 2, …, x N . In three dimensions, we show that, by slightly perturbing f 1, f 2, …, f N , there exists a uniloadable web supporting this loading. Uniloadable means it supports this loading and all positive multiples of it, but not any other loading. Uniloadable webs provide a mechanism for channelling stress in desired ways.

New alternative convex conditions on exponential stability and stabilisation of switched positive linear systems with dwell time

This study is concerned with dwell time stability and stabilisation problems of switched positive linear systems (SPLSs). The dwell time refers to minimum dwell time and constant dwell time. Several stability conditions for primal and transpose SPLSs with dwell time are presented, and the relation between these conditions is illustrated. Some of these conditions are given in terms of infinite-dimensional linear programming (LP), which cannot be solved directly. Then, by utilising the piecewise linear approach, new alternative convex conditions are formulated in terms of finite-dimensional LP. Compared to the existing literature, results with lower or at least the same conservatism can be obtained under the new conditions for the same discretised order. An algorithm is given to reduce the computational cost. Meanwhile, it is proved that there exists a relation between these convex and non-convex conditions if the discretised order is sufficiently large. By utilising the transpose conditions, alternative convex conditions on stabilisation of SPLSs with dwell time are also presented. The controller gain matrices can be computed by solving a set of LP directly. Finally, the correctness and superiority of the results are verified by numerical examples.

An approximate method for solving fractional partial differential equation by using an embedding process

In this article, solving of fractional partial differential equation (FPDE) with optimal time control approach is developed. Conformable derivative as a new definition of fractional derivative is considered. At first the (FPDE) with stationary regime is converted to optimal time control problem, then by using an embedding process, the obtained system is converted to finite-dimensional linear programming (LP) problem and finally optimal time corresponded by (FPDE) system is approximated. For example, this method is used on the conformable fractional heat equation with the initial and boundary conditions.

A Linearization Technique for Solving General 3-D Shape Optimization Problems in Spherical Coordinates

Regarding the some useful advantages of spherical coordinates for some special problems, in this paper, based on Radon measure properties, we present a new and basic solution method for general shape optimization problems defined in spherical coordinates. Indeed, our goal is to determine a bounded shape located over the (x, y)-plane, such that its projection in the (x, y)-plane and its volume is given and also it minimizes some given surface integral. To solve these kinds of problems, we somehow extend the embedding process in Radon measures space. First, the problem is converted into an infinite-dimensional linear programming one. Then, using approximation scheme and a special way for discretization in spherical region, this problem is reduced to a finite-dimensional linear programming one. Finally, the solution of this new problem is used to construct a nearly optimal smooth surface by applying an outlier detection algorithm and curve fitting. More than reducing the complexity, this approach in comparison with the other methods has some other advantages: linear treatment for even nonlinear problems, and the minimization is global and does not depend on initial shape and mesh design. Numerical examples are also given to demonstrate the effectiveness of the new method, especially for classical and obstacle problems.

The Power of Diversity: Data-Driven Robust Predictive Control for Energy-Efficient Buildings and Districts

The cooperative energy management of aggregated buildings has recently received a great deal of interest due to substantial potential energy savings. These gains are mainly obtained in two ways: 1) exploiting the load shifting capabilities of the cooperative buildings and 2) utilizing the expensive but energy-efficient equipment that is commonly shared by the building community (e.g., heat pumps, batteries, and photovoltaics). Several deterministic and stochastic control schemes that strive to realize these savings have been proposed in the literature. A common difficulty with all these methods is integrating knowledge about the disturbances affecting the system. In this context, the underlying disturbance distributions are often poorly characterized based on historical data. In this paper, we address this issue by exploiting the historical data to construct families of distributions, which contain these underlying distributions with high confidence. We then employ tools from data-driven robust optimization to formulate a multistage stochastic optimization problem, which can be approximated by a finite-dimensional linear program. We demonstrate its efficacy in a numerical study, in which it is shown to outperform, in terms of energy cost savings and constraint violations, established solution techniques from the literature. We conclude this paper by showing the significant energy gains that are obtained by cooperatively managing a collection of buildings with heterogeneous characteristics.

Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.

No-arbitrage bounds for the forward smile given marginals

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek [Financ. Stochastics, 2015, 19, 189–214] and by Hobson and Neuberger [Math. Financ., 2012, 22, 31–56]. We recast this dual approach as a finite-dimensional linear program, and reconcile numerically, in the Black–Scholes and in the Heston model, the two approaches.

An approximation scheme for the Kantorovich-Rubinstein problem on compact spaces

AbstractThis paper presents an approximation scheme for the Kantorovich-Rubinstein mass transshipment (KR) problem on compact spaces. A sequence of finite-dimensional linear programs, minimal cost network flow problems with bounds, are introduced and it is proven that the limit of the sequence of the optimal values of these problems is the optimal value of the KR problem. Numerical results are presented approximating the Kantorovich metric between distributions on [0,1].

Extremal Characteristics of Statistical Criteria with Given Total Variation Distances between Hypotheses

We consider extremal values of the characteristics of nonrandomized statistical criteria in the following case: there are $n$ hypotheses with given total variation distances. It is shown that finding the extremal values of a function of error probabilities may be reduced to solving a finite-dimensional linear programming problem. We found exact formulas for extreme values of the sum of all error probabilities in the case n=3.

Interval peak-to-peak observers for continuous- and discrete-time systems with persistent inputs and delays

While the design of optimal peak-to-peak controllers/observers for linear systems is known to be a difficult problem, this problem becomes interestingly much easier in the context of interval observers because of the positive nature of the error dynamics. Indeed, by exploiting several recent results on positive systems, we propose a novel and non-conservative approach formulated in terms of tractable finite-dimensional linear programs for designing a class of interval observers achieving their minimum possible peak-to-peak gain. The optimal observer is notably shown to be uniform over the set of all possible mappings between observation errors and their weighted versions, which parallels a recent result on the stabilization of linear positive systems. Results pertaining on the interval observation of time-delay and discrete-time systems are then obtained as a direct application of the proposed method, emphasizing then its versatility. Several examples on the interval observation of linear and nonlinear systems are finally given for illustration.

Strong duality and sensitivity analysis in semi-infinite linear programming

Finite-dimensional linear programs satisfy strong duality (SD) and have the pricing (DP) property. The DP property ensures that, given a sufficiently small perturbation of the right-hand-side vector, there exists a dual solution that correctly prices the perturbation by computing the exact change in the optimal objective function value. These properties may fail in semi-infinite linear programming where the constraint vector space is infinite dimensional. Unlike the finite-dimensional case, in semi-infinite linear programs the constraint vector space is a modeling choice. We show that, for a sufficiently restricted vector space, both SD and DP always hold, at the cost of restricting the perturbations to that space. The main goal of the paper is to extend this restricted space to the largest possible constraint space where SD and DP hold. Once SD or DP fail for a given constraint space, then these conditions fail for all larger constraint spaces. We give sufficient conditions for when SD and DP hold in an extended constraint space. Our results require the use of linear functionals that are singular or purely finitely additive and thus not representable as finite support vectors. We use the extension of the Fourier---Motzkin elimination procedure to semi-infinite linear systems to understand these linear functionals.