Non-convex Mixed Integer Non-linear Problem Research Articles

Efficient resource allocation for 5G/6G cognitive radio networks using probabilistic interference models

The Cognitive Radio Network, incorporating Device-to-Device communication and a heterogeneous network, has garnered significant attention due to its ability to address spectrum shortage issues and optimize the efficient utilization of spectrum resources. However, resource allocation considering stochastic behavior has not been considered in existing studies. In this paper, our work aimed to maximize the throughput of the overall network considering multiple users under the umbrella of the Cognitive radio network assisted by amplify and forward relay. The constraints are treated as chance constraints with a probability of satisfaction in them, which leads to a non-convex mixed integer nonlinear problem which is an NP-Hard problem. To solve this problem an exhaustive search solution for optimal results is required. However, the computational burden always increases with the user equipment. Therefore, to obtain an optimal solution with having low computational burden, the Outer Approximation Algorithm is utilized in this research. To evaluate the desired results, extensive simulations have been carried out. The effectiveness of the proposed algorithm is verified by results in terms of throughput maximization under the impact of chance constraint formulation in the cognitive radio networks.

Task Offloading in Multi-Access Edge Computing Enabled UAV-Aided Emergency Response Operations

In emergency response operations, using uncrewed aerial vehicles (UAVs) has recently become a promising solution due to their flexibility and easy deployment. However, tasks performed by the UAVs, e.g., object detection and human pose recognition, usually require a high computation capacity and energy supply. Furthermore, offloading tasks to the edge server-equipped base stations may not always be possible because of a lack of infrastructure or distance. Therefore, UAV-aided edge servers can be deployed near UAV scouts to provide computing services. However, a UAV can not perform all types of tasks since it has limitations on memory, available software, central processing unit (CPU), and graphics processing unit (GPU) capacity. Therefore, this study focuses on task offloading (TO), power, and computation resource allocation (PRA) problems in a multi-layer MEC-enabled UAV network while taking into account CPU and GPU requirements of tasks, the capacity of the devices (i.e., computational resources, power, and energy), and limitations on the type of tasks a UAVs can perform. The problem is formulated as a non-convex mixed-integer nonlinear problem to minimize the weighted sum of the maximum energy consumption ratio in the network and total task execution latency ratio, and then decomposed and converted into an integer and a convex problem. A messy genetic algorithm (mGA)-based TO and PRA strategy (mGA-TPR) is proposed to solve the problem, where two PRA strategies are based on the Karush–Kuhn–Tucker conditions used to solve the PRA problem. Simulation results verify that the proposed scheme can outperform the baseline methods.

Reinforcement-Learning-Based Resource Allocation for Energy-Harvesting-Aided D2D Communications in IoT Networks

This article proposes a novel approach to improve the energy efficiency (EE) of an energy-harvesting (EH)-enabled IoT network supported by simultaneous wireless information and power transfer (SWIPT). More specifically, the device-to-device (D2D) users harvest ambient energy throughout their communication with the time switching (TS) technique, while the Internet of Things (IoT) users harvest energy from the base station (BS) based on the power splitting (PS) method. We study the EE optimization problem that takes into account transmit power feasibility conditions for D2D users and IoT users, the minimum data rate requirements for D2D and IoT users, joint spectrum sharing block, and time allocation for the D2D links. The underlying problem is a highly nonconvex mixed-integer nonlinear problem (MINLP) and the global optimal solution is intractable. To handle it, we decompose the original problem into three subproblems: 1) joint subchannel allocation and PS; 2) power control; and 3) time allocation. Since it is difficult to find an exact state model approach in a dynamic environment with a large state space, we exploit a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> -learning method based on reinforcement learning (RL) to solve the first subproblem. To solve the second subproblem, we apply a conventional convex optimization technique based on the majorization–minimization (MM) approach and Dinkelbach method. Simulation results not only demonstrate the superiority of our proposed algorithm as compared to other methods in the literature but also confirm impressive EE gains through spectrum sharing and harvested energy from D2D and IoT users.

Global optimization for the multilevel European gas market system with nonlinear flow models on trees

The European gas market is implemented as an entry-exit system, which aims to decouple transport and trading of gas. It has been modeled in the literature as a multilevel problem, which contains a nonlinear flow model of gas physics. Besides the multilevel structure and the nonlinear flow model, the computation of so-called technical capacities is another major challenge. These lead to nonlinear adjustable robust constraints that are computationally intractable in general. We provide techniques to equivalently reformulate these nonlinear adjustable constraints as finitely many convex constraints including integer variables in the case that the underlying network is tree-shaped. We further derive additional combinatorial constraints that significantly speed up the solution process. Using our results, we can recast the multilevel model as a single-level nonconvex mixed-integer nonlinear problem, which we then solve on a real-world network, namely the Greek gas network, to global optimality. Overall, this is the first time that the considered multilevel entry-exit system can be solved for a real-world sized network and a nonlinear flow model.

Energy Efficiency Optimization for Massive MIMO Backhaul Networks with Imperfect CSI and Full Duplex Small Cells

In this paper, a massive MIMO enabled backhaul model for two-tier heterogeneous networks with in-band full-duplex transmission and imperfect CSI is provided. Because of a massive number of antennas at the macro base station and densification of small cells, circuit power consumption increases in the system. It is requisite to study energy efficiency of such backhaul networks. This paper aims at maximizing EE for optimal user association, spectrum allocation, and power allocation while accounting for QoS and backhaul constraints. A joint optimization problem is formulated as a non-convex mixed integer non-linear problem which provides an un-affordable computational complexity. To crack the problem efficiently, it is progressively divided into sub-problems until each one is convex and is solved separately to obtain the optimal solution. Furthermore, complexity of the proposed algorithm is evaluated and Jain’s fairness index is measured. Simulation results of the proposed distributive algorithm are shown and compared with that of the existing algorithm (Max SINR) to manifest the strength of the proposed algorithm. The impact of self-interference cancellation factor and channel estimation error on EE is clearly reflected by the numerical results. In addition, it is observed that using the proposed algorithm results in better EE than increasing the number of antennas.

Efficient resource allocation algorithms for high energy efficiency with fairness among users in OFDMA networks

In this article, we devise two efficient radio resource allocation algorithms in order to increase fair energy efficiency among users of OFDMA networks. The objective function is considered to be the well-known max-min function in order for allocating radio resources fairly in terms of energy efficiency, by taking the maximum transmit power, minimum rate requirements, and subchannel assignment constraints into account. The resulting problem is a nonconvex mixed-integer nonlinear problem for which two algorithms are proposed. In the first proposed algorithm, by reallocating the transmit power on subchannels, we obtain a suitable subchannel assignment with high fair energy efficiency. In the second proposed algorithm, by applying generalized fractional programming, mathematically modifying the objective function and constraints to a continuous and convex form and employing sequential convex programming, we allocate power and subchannel jointly. The proposed algorithms efficiently utilize radio resources and therefore, yield a far better performance than available solutions.

A Hybrid Optimization Approach for Residential Energy Management

In the past few years, the development of demand response (DR) programs for smart grid systems has provided residential customers with a real opportunity to participate in DR-driven projects. One typical DR-related application now available is automated energy scheduling for residential building, which can be used to help reduce energy costs. Residential energy scheduling focuses on saving cost by managing the operation time and energy consumption level of different appliances. Demand Response programs present as NP-hard problems, the equations are non-convex mixed integer non-linear problems (MINLP), for which it is difficult to obtain an optimal solution. In relation to residential DR, we propose a hybrid approach here that is able to solve an MINLP. The problem is decomposed into discrete and continuous variables. The discrete variables are optimized by using a particle swarm optimization (PSO) algorithm, whilst the continuous variables are determined by using a gradient-based deterministic algorithm. The superiority of the proposed algorithm was demonstrated by compared with commercial optimization software and heuristic based algorithm. Furthermore, the effectiveness of the proposed method in residential DR was tested and the results were presented.

Maximizing the storage capacity of gas networks: a global MINLP approach

In this paper, we study the transient optimization of gas networks, focusing in particular on maximizing the storage capacity of the network. We include nonlinear gas physics and active elements such as valves and compressors, which due to their switching lead to discrete decisions. The former is described by a model derived from the Euler equations that is given by a coupled system of nonlinear parabolic partial differential equations ( $${\text{PDEs}}$$ ). We tackle the resulting mathematical optimization problem by a first-discretize-then-optimize approach. To this end, we introduce a new discretization of the underlying system of parabolic $${\text{PDEs}}$$ and prove well-posedness for the resulting nonlinear discretized system. Endowed with this discretization, we model the problem of maximizing the storage capacity as a non-convex mixed-integer nonlinear problem ( $${\text{MINLP}}$$ ). For the numerical solution of the $${\text{MINLP}}$$ , we algorithmically extend a well-known relaxation approach that has already been used very successfully in the field of stationary gas network optimization. This method allows us to solve the problem to global optimality by iteratively solving a series of mixed-integer problems. Finally, we present two case studies that illustrate the applicability of our approach.

Strengthening the sequential convex MINLP technique by perspective reformulations

The sequential convex MINLP (SC-MINLP) technique is a solution method for nonconvex mixed-integer nonlinear problems (MINLPs) where the nonconvexities are separable. It is based on solving a sequence of convex MINLPs which trade a better and better relaxation of the nonconvex part of the problem with the introduction of more and more piecewise-linear nonconvex terms, and therefore binary variables. The convex MINLPs are obtained by partitioning the domain of each separable nonconvex term in the intervals in which it is convex and those in which it is concave. In the former, the term is left in its original form, while in the latter it is piecewise-linearized. Since each interval corresponds to a semi-continuous variable, we propose to modify the convex terms using the Perspective Reformulation technique to strengthen the bounds. We show by means of experimental results on different classes of instances that doing so significantly decreases the solution time of the convex MINLPs, which is the most time consuming part of the approach, and has therefore the potential to improving the overall effectiveness of SC-MINLP.

On the statistical evaluation of algorithmic's computational experimentation with infeasible solutions

The experimental evaluation of algorithms results in a large set of data which generally do not follow a normal distribution or are not heteroscedastic. Besides, some of its entries may be missing, due to the inability of an algorithm to find a feasible solution until a time limit is met. Those characteristics restrict the statistical evaluation of computational experiments. This work proposes a bi-objective lexicographical ranking scheme to evaluate datasets with such characteristics. The output ranking can be used as input to any desired statistical test. We used the proposed ranking scheme to assess the results obtained by the Iterative Rounding heuristic (IR). A Friedman's test and a subsequent post-hoc test carried out on the ranked data demonstrated that IR performed significantly better than the Feasibility Pump heuristic when solving 152 benchmark problems of Nonconvex Mixed-Integer Nonlinear Problems. However, is also showed that the RECIPE heuristic was significantly better than IR when solving the same benchmark problems.

A model for multi-class road network recovery scheduling of regional road networks

In this paper, an optimisation model for recovery planning of road networks is presented in which both social and economic resilience is aimed to be achieved. The model is formulated as a bi-level multi-objective discrete network design problem which forms a non-convex mixed integer non-linear problem. Solved by a Branch and Bound method, the solution algorithm employs an outer approximation method to estimate the lower bound of each node in the Branch and Bound search tree. The solution algorithm exploits a unique approach for lower-bound computation dealing with a disrupted multi-class network that may not be able to satisfy the demand between all OD pairs due to damaged links. The model is assessed by applying it on the Sioux Falls network. It is also illustrated how the Pareto-optimal solutions achieved by the multi-objective optimisation can vary depending on the emphasis placed on different classes of vehicles.

Nonconvex optimization of desirability functions

ABSTRACTDesirability functions (DFs) are commonly used in optimization of design parameters with multiple quality characteristic to obtain a good compromise among predicted response models obtained from experimental designs. Besides discussing multi-objective approaches for optimization of DFs, we present a brief review of literature about most commonly used Derringer and Suich type of DFs and others as well as their capabilities and limitations. Optimization of DFs of Derringer and Suich is a challenging problem. Although they have an advantageous shape over other DFs, their nonsmooth nature is a drawback. Commercially available software products used by quality engineers usually do optimization of these functions by derivative free search methods on the design domain (such as Design-Expert), which involves the risk of not finding the global optimum in a reasonable time. Use of gradient-based methods (as in MINITAB) after smoothing nondifferentiable points is also proposed as well as different metaheuristics and interactive multi-objective approaches, which have their own drawbacks. In this study, by utilizing a reformulation on DFs, it is shown that the nonsmooth optimization problem becomes a nonconvex mixed-integer nonlinear problem. Then, a continuous relaxation of this problem can be solved with nonconvex and global optimization approaches supported by widely available software programs. We demonstrate our findings on two well-known examples from the quality engineering literature and their extensions.

Joint Sub-Channel Assignment and Power Allocation in Heterogeneous Networks: An Efficient Optimization Method

This letter is concerned with joint sub-channel and power allocation in a heterogeneous network. We consider the uplink of a co-channel framework where a macrocell shares the orthogonal frequency division multiple access (OFDMA) sub-channels with a number of small cells (SCs). We maximize the weighted sum-rate of the OFDMA SC network, while a target constraint on the cross-tier interference and a minimum data rate requirement for the high data rate SC users are satisfied. The optimization problem is a highly non-convex mixed integer non-linear problem. We first reformulate the problem into a more mathematically tractable form. Then, we relax the integer variables by adding a penalty term to the objective function. Finally, we rewrite the objective function as the difference of two concave (D.C.) functions and solve this problem using the successive convex approximation method. We show that our method outperforms the state-of-the-art schemes.

Optimal TXOP Sharing in IEEE 802.11ac

With the goal of throughput maximization in IEEE 802.11ac, this letter proposes a throughput-aware TXOP sharing scheme. This scheme performs joint link adaptation and scheduling of the downlink multiuser transmission such that MAC layer throughput is maximized, while the TXOP sharing rules of IEEE 802.11ac are satisfied. We show that this joint optimization problem is a non-convex mixed-integer nonlinear problem and solve it employing an ant colony optimization-based metaheuristic. The simulation results generated by IEEE 802.11ac channel and physical layer models substantiate a significant gain of this scheme in terms of MAC layer throughput.

Joint Resource Allocation for Max-Min Throughput in Multicell Networks

We investigate the resource-allocation problem in multicell networks targeting the max-min throughput of all cells. A joint optimization over power control, channel allocation, and user association is considered, and the problem is then formulated as a nonconvex mixed-integer nonlinear problem (MINLP). To solve this problem, we proposed an alternating-optimization-based algorithm, which applies branch-and-bound and simulated annealing in solving subproblems at each optimization step. We also demonstrate the convergence and efficiency of the proposed algorithms by thorough numerical experiments. The experimental results show that joint optimization over all resources outperforms the restricted optimization over individual resources significantly.

Optimizing the design of complex energy conversion systems by Branch and Cut

The paper examines the applicability of mathematical programming methods to the simultaneous optimization of the structure and the operational parameters of a combined-cycle-based cogeneration plant. The optimization problem is formulated as a nonconvex mixed-integer nonlinear problem (MINLP) and solved by the MINLP solver LaGO. The algorithm generates a convex relaxation of the MINLP and applies a Branch and Cut algorithm to the relaxation. Numerical results for different demands for electric power and process steam are discussed and a sensitivity analysis is performed.

Improving Mathematical Optimization Techniques with the Aid of Exergy-Based Variables

The design optimization of energy conversion plants requires sophisticated optimization techniques. The usefulness of mathematical programming approaches has been discussed in several papers. Usually, the quality of the computed solutions, concerning global optimality and the convergence speed, is not discussed in these papers and even the existence of local optimal solutions is not mentioned. Indeed, the optimization of nonconvex mixed integer non-linear problems (MINLP), such as the structural and design optimization of power plants, is a very difficult problem. However, knowledge of the real optimization potential can assist the design engineer in better understanding the optimization procedure. This article deals with the use of exergetic variables for improving the quality of results obtained from mathematical optimization techniques and their convergence speed. LaGO, the solver used to compute the discussed results, can evaluate the obtained solution of the discussed minimization problems by calculating lower bounds of the original problem based on a relaxed convex objective function. Here, the use of exergetic variables can help to increase the lower bounds significantly and thus, to improve the evaluation of the computed solutions and the convergence speed. The method is applied to different optimization tasks. This paper is an updated version of a paper published in the ECOS'08 proceedings.

Optimal design of reverse-osmosis networks for wastewater treatment

An optimization study of reverse-osmosis networks (RON) for wastewater treatment has been carried out by describing the system as a nonconvex mixed-integer nonlinear problem (MINLP). A mixed-integer linear problem (MILP) is derived from the original nonlinear problem by the convex relaxation of the nonconvex terms in the MINLP to provide bounds for the global optimum. The MILP model is solved iteratively to supply different initial guesses for the nonconvex MINLP model. It is found that such a procedure is effective in finding local optimum solutions in reasonable time and overcoming possible convergence difficulties associated with MINLP local search methods. Examples of water desalination and wastewater treatment from the pulp and paper industry are considered as case studies to illustrate the proposed solution strategy.