Minimax Objective Function Research Articles

Data-driven robust model predictive control for greenhouse temperature control and energy utilisation assessment

The greenhouse microclimate, especially temperature, is highly complex, and controlling it requires significant resources due to the greenhouses' inefficient design. The application of model predictive control is a promising strategy for temperature control and efficient greenhouse management. However, it does not account for the inaccuracies and uncertainties existing in the system, leading to sub-optimal temperatures. Therefore, this study proposes a comprehensive data-driven robust model predictive control framework for greenhouse temperature control and its energy utilisation assessment in the presence of uncertainties. First, an analytical model based on mass and energy balance and a data-driven model based on an artificial neural network is developed, and their prediction performance is compared. The artificial neural network demonstrates a higher prediction accuracy and is used as the system model in the proposed control framework. A robust model predictive control strategy, based on the minimax objective function and particle swarm optimisation algorithm, is developed to handle the uncertainties in the system. Results illustrate that in the presence of uncertainties, the robust model predictive control strategy outperforms the existing greenhouse climate management system and basic model predictive control with an RMSE of 0.32 °C and 0.60 °C for a two-day simulation period in winter and summer, respectively. Furthermore, the robust model predictive control strategy leads to an energy reduction of 9.67% and 23.61% in winter and summer. The proposed framework is flexible and general and can be applied to other greenhouses with different configurations and cultivated crops by fine-tuning it on the new data set.

Convergence of Non-Convex Non-Concave GANs Using Sinkhorn Divergence

Sinkhorn divergence is a symmetric normalization of entropic regularized optimal transport. It is a smooth and continuous metrized weak-convergence with excellent geometric properties. We use it as an alternative for the minimax objective function in formulating generative adversarial networks. The optimization is defined with Sinkhorn divergence as the objective, under the non-convex and non-concave condition. This work focuses on the optimization’s convergence and stability. We propose a first order sequential stochastic gradient descent ascent (SeqSGDA) algorithm. Under some mild approximations, the learning converges to local minimax points. Using the structural similarity index measure (SSIM), we supply a non-asymptotic analysis of the algorithm’s convergence rate. Empirical evidences show a convergence rate, which is inversely proportional to the number of iterations, when tested on tiny colour datasets Cats and CelebA on the deep convolutional generative adversarial networks and ResNet neural architectures. The entropy regularization parameter $\varepsilon $ is approximated to the SSIM tolerance $\epsilon $ . We determine that the iteration complexity to return to an $\epsilon $ -stationary point to be $\mathcal {O}\left ({\kappa \, \log (\epsilon ^{-1})}\right)$ , where $\kappa $ is a value that depends on the Sinkhorn divergence’s convexity and the minimax step ratio in the SeqSGDA algorithm.

Functional gradient approach to probabilistic minimax active learning

Many active learning methods select informative and representative examples by employing a parametric approach. However, there is very limited research pursuing a functional non-parametric approach to informative and representative active learning. The present study proposes a general functional approach to active learning based on a minimax objective function. Using this general algorithm, the current paper presents a specific algorithm based on a simple class of functions. Experiments show that the proposed method is efficient in selecting examples. It is interesting that the resulting algorithm can be interpreted from a spectral filtering perspective. This establishes a relationship between active learning, boosting, and spectral filtering and opens up new avenues for developing even better active learning algorithms.

Optimal Operation of Interprovincial Hydropower System Including Xiluodu and Local Plants in Multiple Recipient Regions

This paper focuses on the monthly operations of an interprovincial hydropower system (IHS) connected by ultrahigh voltage direct current lines. The IHS consists of the Xiluodu Hydropower Project, which ranks second in China, and local plants in multiple recipient regions. It simultaneously provides electricity for Zhejiang and Guangdong provinces and thus meets their complex operation requirements. This paper develops a multi-objective optimization model of maximizing the minimum of total hydropower generation for each provincial power grid while considering network security constraints, electricity contracts, and plant constraints. The purpose is to enhance the minimum power in dry season by using the differences in hydrology and regulating storage of multiple rivers. The TOPSIS method is utilized to handle this multi-objective optimization, where the complex minimax objective function is transformed into a group of easily solved linear formulations. Nonlinearities of the hydropower system are approximatively described as polynomial formulations. The model was used to solve the problem using mixed integer nonlinear programming that is based on the branch-and-bound technique. The proposed method was applied to the monthly generation scheduling of the IHS. Compared to the conventional method, both the total electricity for Guangdong Power Grid and Zhejiang Power Grid during dry season increased by 6% and 4%, respectively. The minimum monthly power also showed a significant increase of 40% and 31%. It was demonstrated that the hydrological differences between Xiluodu Plant and local hydropower plants in receiving power grids can be fully used to improve monthly hydropower generation.

Using Mathematical Programming to Select and Seed Teams for the NCAA Tournament

We develop a mathematical programming model to select, rank, and seed 68 teams for the National Collegiate Athletic Association (NCAA) Men’s Basketball Tournament. The selections and seeding are the responsibility of a 10-person selection committee, which chooses from 351 Division I college basketball teams and considers many team-performance attributes. Our approach yields a logically consistent initial starting point based on finding dominant relationships among the teams, and the minimax objective function. We first apply the procedure to data for each of the five tournaments from 2012 to 2016 and show that our rankings align well with those of the respective committees. We then implement the procedure and obtain a set of recommendations for the 2017 NCAA tournament’s committee, prior to the announcement of its selections. Our model’s recommendations closely align with those of the committee, thus supporting our contention that the approach provides a viable, readily implemented, and easily understood basis for both informing the committee’s decisions and encouraging its members to articulate their reasons for choosing otherwise. The approach requires no training data from prior seasons or previous recommendations, and can be expanded to other applications where decision makers are tasked with selecting and ranking a set of entities that has quantifiable performance attributes.

A stochastic algorithm for makespan minimized multi-agent path planning in discrete space

Makespan minimized multi-agent path planning (MAPP) requires the minimization of the time taken by the slowest agents to reach its destination. The resulting minimax objective function is non-smooth and the search for an optimal solution in MAPP can be intractable. In this work, a maximum entropy function is adopted to approximate the minimax objective function. An iterative algorithm named probabilistic iterative makespan minimization (PIMM) is then proposed to approximate a makespan minimized MAPP solution by solving a sequence of computationally hard MAPP minimization problems with a linear objective function. At each iteration, a novel local search algorithm called probabilistic iterative path coordination (PIPC) is used to find a sufficiently good solution for each MAPP minimization problem. Experimental results from comparative studies with existing MAPP algorithms show that the proposed algorithm strikes a good tradeoff between the quality of the makespan minimized solution and the computational cost incurred.

Conflicting Congestion Effects in Resource Allocation Games

We study strategic resource allocation settings, where jobs correspond to self-interested players who choose resources with the objective of minimizing their individual cost. Our framework departs from the existing game-theoretic models mainly in assuming conflicting congestion effects, but also in assuming an unlimited supply of resources. In our model, a job's cost is composed of both its resource's load (which increases with congestion) and its share in the resource's activation cost (which decreases with congestion). We provide results for a job-scheduling setting with heterogeneous jobs and identical machines. We show that if the resource's activation cost is shared equally among its users, a pure Nash equilibrium (NE) might not exist. In contrast, the proportional sharing rule induces a game that admits a pure NE, which can also be computed in polynomial time. As part of the algorithm's analysis, we establish a new, nontrivial property of schedules obtained by the longest processing time algorithm. We also observe that, unlike in congestion games, best-response dynamics (BRD) are not guaranteed to converge to a Nash equilibrium. Finally, we measure the inefficiency of equilibria with respect to the minimax objective function, and prove that there is no universal bound for the worst-case inefficiency (as quantified by the “price of anarchy” measure). However, the best-case inefficiency (quantified by the “price of stability” measure) is bounded by 5/4, and this is tight. These results add another layer to the growing literature on the price of anarchy and stability, which studies the extent to which selfish behavior affects system efficiency.

Optimal design of vibration absorber using minimax criterion with simplified constraints

In this paper, a minimax design of damped dynamic vibration absorber for a damped primary system is investigated to minimize the vibration magnitude peaks. Moreover, to reduce the sensitivity of the primary system response to variations of the forcing frequency for a two-degree-of-freedom system, the primary system should have two equal resonance magnitude peaks. To meet this requirement, a set of simplified constraint equations including distribution characteristics of the resonant frequencies of the primary system is established for the minimax objective function. The modified constraint equations have less unknown variables than those by other authors, which not only simplifies the computation but also improves the accuracy of the optimal values. The advantage of the proposed method is illustrated through numerical simulations.

An Equitable Bandwidth Allocation Model for Video-on-Demand Networks

Video-on-Demand (VOD) or near-VOD services are expected to grow significantly over time, providing diverse programs for home entertainment, learning and training, news-on-demand, and other applications. These services require large bandwidth resources. We present a model for bandwidth allocation in a tree network with limited link capacities, where a server at the root node repeatedly broadcasts copies of various programs. The time intervals between successive broadcasts of each program can be increased at subsequent nodes, or the video quality can be decreased, thus providing different service performance to different nodes while satisfying the capacity constraints. The model is formulated as an equitable resource allocation problem with a lexicographic minimax objective function and tree-like ordering constraints. We present a lexicographic minimax algorithm that allocates each link’s bandwidth among the programs carried on the link. The algorithm repeatedly solves minimax problems, and fixes some variables at their optimal value after the solution of each such problem. The algorithm for solving the minimax problems uses a bisection search to find the minimax solution with the minimal decision variable values. The model also provides an ordered list of links from the most critical link to the least critical link, a useful feature for capacity expansion planning decisions.

A solution to the hub center problem via a single-relocation algorithm with tabu search

Network designers for many transportation or telecommunication applications need to minimize the maximum (weighted) interaction time or distance in the networks, i.e., to solve the p-hub center problems. This study addresses the p-hub center problem. A single-relocation heuristic is proposed as a means of generating location-allocation strategies in a reasonable amount of time, and tabu search is superimposed on the underlying algorithm, so as to decrease the possibility of being trapped by local optima. Two single-allocation schemes are adapted from the literature, and a further one is developed specifically for the minimax objective function, for use in the evaluation phase of the algorithm. A greedy local search is employed to improve the resulting allocations. A number of strategies for choosing the initial hub locations are discussed and initial strategy - allocation scheme combinations are compared in terms of solution quality and pragmatic efficiency.

A solution to the hub center problem via a single-relocation algorithm with tabu search

Network designers for many transportation or telecommunication applications need to minimize the maximum (weighted) interaction time or distance in the networks, i.e., to solve the p-hub center problems. This study addresses the p-hub center problem. A single-relocation heuristic is proposed as a means of generating location-allocation strategies in a reasonable amount of time, and tabu search is superimposed on the underlying algorithm, so as to decrease the possibility of being trapped by local optima. Two single-allocation schemes are adapted from the literature, and a further one is developed specifically for the minimax objective function, for use in the evaluation phase of the algorithm. A greedy local search is employed to improve the resulting allocations. A number of strategies for choosing the initial hub locations are discussed and initial strategy - allocation scheme combinations are compared in terms of solution quality and pragmatic efficiency.

A Resource Allocation Approach for the Generation of Service-Dependent Demand Matrices for Communications Networks

Estimating point-to-point demands from partially available information, such as total demand volumes originating and terminating at nodes and traffic volumes routed on links, has significant applications in various areas, such as communications network planning and transportation planning. Existing methods include matrix and link scaling methods, statistical methods, more complex mathematical programming models, and forecasting using demographic data. We present a new mathematical programming model based on equitable resource allocation. The model considers multiple services, e.g., data, video, and voice, and generates a point-to-point demand matrix for each service. Originating and terminating demands for each service and link loads, aggregated over all services, are viewed as resources. Each point-to-point demand is associated with a performance function that measures its weighted, normalized deviation from a target defined by a service-dependent community of interest matrix. The model formulation has a lexicographic minimax objective function and multiple knapsack resource constraints. The model has an intuitively appealing interpretation and a specialized algorithm can generate demand matrices for large network problems very fast.

Linear Formulation of a Distributed Boundary Control Problem

In this paper, we consider a distributed boundary control problem governed by an elliptic partial differential equation with state constraints and a minimax objective function. The continuous optimal control problem, discretized with the finite element method, is numerically approximated by a family of linear programming problems. Application to an optimal configuration problem is discussed.

Modeling uncertainty and analyzing bottleneck characteristics in multiperiod design optimization

In this paper we address modeling, analysis and computational synthesis issues in multipenod design problems. We present a novel minimax approach for the formulation of the objective function, to address uncertainty in the duration of individual periods. For the analysis of these problems, we offer a formal definition of bottleneck periods and derive properties that contribute to the design process. At the level of synthesis and based on the bottleneck properties, we propose a decomposition strategy for general MINLP multiperiod problems where the MILP part does not scale with the total number of periods. As shown through the derived properties and a set of example problems, the minimax objective function yields designs with enhanced robustness properties compared to the standard multiperiod objective. The bottleneck period concepts provide an analysis framework that yields a better under—standing of multiperiod problems and provides guidelines for an improved design procedure. Finally, as predicted by the theoretical derivation and illustrated through examples, the proposed bottleneck-period based OA/BP method is computationally superior to both Outer-Approximation (OA) and Generalized Benders Decomposition (GBD).

Reducibility of minimax to minisum 0–1 programming problems

We consider 0–1 programming problems with a minimax objective function and any set of constraints. Upon appropriate transformations of its cost coefficients, such a minimax problem can be reduced to a linear minisum problem with the same set of feasible solutions such that an optimal solution to the latter will also solve the original minimax problem. Although this reducibility applies for any 0–1 programming problem, it is of particular interest for certain locational decision models. Among the obvious implications are that an algorithm for solving a p-median (minisum) problem in a network will also solve a corresponding p-center (minimax) problem. It should be emphasized that the results presented will in general only hold for 0–1 problems due to intrinsic properties of the minimax criterion.

A Projected Lagrangian Algorithm for Nonlinear Minimax Optimization

The minimax problem is an unconstrained optimization problem whose objective function is not differentiable everywhere, and hence cannot be solved efficiently by standard techniques for unconstrained optimization. It is well known that the problem can be transformed into a nonlinearly constrained optimization problem with one extra variable, where the objective and constraint functions are continuously differentiable. This equivalent problem has special properties which are ignored if solved by a general-purpose constrained optimization method. The algorithm we present exploits the special structure of the equivalent problem. A direction of search is obtained at each iteration of the algorithm by solving an equality-constrained quadratic programming problem, related to one a projected Lagrangian method might use to solve the equivalent constrained optimization problem. Special Lagrange multiplier estimates are used to form an approximation to the Hessian of the Lagrangian function, which appears in the quadratic program. Analytical Hessians, finite differencing or quasi-Newton updating may be used in the approximation of this matrix. The resulting direction of search is guaranteed to be a descent direction for the minimax objective function. Under mild conditions the algorithms are locally quadratically convergent if analytical Hessians are used.

Linearly constrained minimax optimization

We present an algorithm for nonlinear minimax optimization subject to linear equality and inequality constraints which requires first order partial derivatives. The algorithm is based on successive linear approximations to the functions defining the problem. The resulting linear subproblems are solved in the minimax sense subject to the linear constraints. This ensures a feasible-point algorithm. Further, we introduce local bounds on the solutions of the linear subproblems, the bounds being adjusted automatically, depending on the quality of the linear approximations. It is proved that the algorithm will always converge to the set of stationary points of the problem, a stationary point being defined in terms of the generalized gradients of the minimax objective function. It is further proved that, under mild regularity conditions, the algorithm is identical to a quadratically convergent Newton iteration in its final stages. We demonstrate the performance of the algorithm by solving a number of numerical examples with up to 50 variables, 163 functions, and 25 constraints. We have also implemented a version of the algorithm which is particularly suited for the solution of restricted approximation problems.

Wave Sensitivities of Networks

A theoretical foundation is presented for the efficient computation of first- and second-order sensitivities of networks with respect to network parameters in terms of wave variables. The concept of the adjoint network is used. First-order sensitivity formulas for a wide variety of elements are presented, including lumped and uniformly distributed elements, active and passive elements, and reciprocal and nonreciprocal elements. Parameters include electrical quantities, geometrical dimensions, and frequency. It is shown how gradients related to wave-based least pth and minimax objective functions can be computed. A comparison with a method which avoids the need for analysis of adjoint networks is made. Applications in the computer-aided design of networks using efficient gradient minimization methods are envisaged.

Optimal control trajectories with minimax objective functions by linear programming

A standard linear programming code may be used to compute optimal trajectories for a linear discrete-time system with respect to a minimax criterion on either state or control trajectories. Arbitrary linear constraints, equality or inequality, constant or time-varying, may be placed on linear combinations of the state or control variables along their trajectories or at a fixed terminal time. An important feature of the method is that the dimension of the linear programming problem is independent of the dimension of the state space but depends entirely on the numbers of control variables, constraints, and time intervals. Optimal trajectories for a 21st- order system have been calculated in a few minutes of computer time.

Optimal control trajectories with minimax objective functions by linear programming

Optimal control trajectories with minimax objective functions by linear programming