Outer Approximation Algorithm Research Articles

Resource allocation for licensed and unlicensed spectrum in 5G heterogeneous networks

AbstractThe foresighted enormous increase in mobile traffic over the coming years will result in severe congestion on available radio spectrum. The use of additional spectrum in future fifth‐generation mobile networks will be inevitable. Long‐Term Evolution Unlicensed (LTE‐U) is an evolving technology, which effectively utilizes available unlicensed spectrum to increase the capacity of unified fifth‐generation network. However, LTE‐U causes severe interference to existing WiFi networks, which needs to be addressed to take full advantage of LTE‐U. In this paper, we thrive to provide a suboptimal resource allocation for coexisting LTE‐U and WiFi networks to maximize throughput and, hence, minimize the interference. We formulated an optimization problem for joint user association and power allocation for licensed and unlicensed spectrum with objective to maximize sum rate of LTE‐U/WiFi heterogeneous network in a multioperator scenario, subject to minimum rate guarantee and cochannel interference threshold. We propose mesh adaptive direct search algorithm as solution to optimization problem to obtain ϵ‐optimal results. The performance of proposed algorithm is shown in terms of network key performance indicators such as throughput and number of users accommodated. We also benchmark the results from mesh adaptive direct search against outer approximation algorithm.

Approximation of convex bodies by multiple objective optimization and an application in reachable sets

In this paper, we focus on approximating convex compact bodies. For a convex body described as the feasible set in objective space of a multiple objective programme, we show that finding it is equivalent to finding the non-dominated set of a multiple objective programme. This equivalence implies that convex bodies can be approximated using multiple objective optimization algorithms. Therefore, we propose a revised outer approximation algorithm for convex multiple objective programming problems to approximate convex bodies. Finally, we apply the algorithm to solve reachable sets of control systems and use numerical examples to show the effectiveness of the algorithm.

Outer approximation algorithms to solve semi-infinite environmental pollution problems

Introduction: The traditional approaches of managing the integral estimation of a general function do no solve the problema of fulfilling the constraints in each point of the domain. This is achieved through a semi-infinite formulation of the problem, in which there are great possibilities in modeling many practical situations of industrial ecology. Some of them are presented in this paper.Objective: Based on detailed studies ([1]-[3]), the aim of this research paper is to examine the mentioned studies in order to contribute to disseminate SIP (Semi-Infinite Programming) applications in environmental pollution control.Methodology: With regard to the numerical solution a stochastic outer approximation algorithm was used, in addition numerical experiments were performed in MATLAB.Results: The obtained results of this work have showed three patterns, the first one enhances the power of emitting pollution sources, taking into account the area landscape, the second one also enhances the power of pollution sources using wind displacement of emissions, finally the third pattern operates by improving the pollution of 3D areas between a group of emission sources and purifiers.Conclusions: In conclusion, it can be said that the issue of power from emitting pollution sources, by considering the area landscape and wind displacement of emissions, should be modeled such as a semi-infinite programming trouble. The method used to calculate the algorithm stopping criterion, does not interpret this solution as an additional problem.

A numerical method for the approximation of reachable sets of linear control systems

Abstract In this article, a numerical method for the approximation of reachable sets of linear control systems is discussed. First a continuous system is transformed into a discrete one with Runge–Kutta methods. Then based on Benson’s outer approximation algorithm for solving multiobjective optimization problems, we propose a variant of Benson’s algorithm to sandwich the reachable set of the discrete system with an inner approximation and an outer approximation. By specifying an approximation error, the quality of the approximations measured in Hausdorff distance can be directly controlled. Furthermore, we use an illustrative example to demonstrate the working of the algorithm. Finally, computational experiments illustrate the superior performance of our proposed algorithm compared to a recent algorithm in the literature.

Resource Allocation for Energy Harvesting-Powered D2D Communication Underlaying Cellular Networks

Energy harvesting (EH)-powered wireless communications have attracted great attention from both industry and academia. However, the available energy, which relies on EH efficiency, will become another nonnegligible factor when we do research on EH-powered wireless communications. This paper studies the resource allocation problems in terms of spectrum and energy under EH-powered Device-to-Device (D2D) Communication underlaying Cellular Network (EH-DCCN). In this network, D2D pairs powered by EH are allowed to reuse the spectrum resources occupied by Cellular Users (CUs). To investigate the resource allocation problems, a sum-rate maximization problem of the whole cellular network with consideration of Quality of Service (QoS) and available energy constraints is formulated. The maximization problem is a nonconcave mixed-integer nonlinear programming (MINLP) problem, which has been proved to be NP-hard. To solve the problem, we first relax it with a concave lower bound on the original problem and then obtain the theoretical performance of the lower bound by Outer Approximation Algorithm. Moreover, a heuristic algorithm, an Energy-aware Space Matching approach (ESM), is proposed to acquire a suboptimal solution with low computational complexity. Finally, numerical simulation results indicate our considered resource allocation strategy is more effective than the strategy only based on channel state information under the EH-DCCN. Moreover, the performance in aspects of the sum rate and the matching probability shows that the ESM can approximately obtain the theoretical performance of the lower bound on the original problem under the scenarios with higher ratio of CU and EH-powered D2D numbers.

Combinatorial resource allocation in D2D assisted heterogeneous relay networks

Exponential increase in wireless capacity demands innovative network designs. Heterogeneous networks (HetNets) have been considered as a promising solution to ever increasing wireless capacity demand and efficient spectrum utilization. However severe interference scenarios exist in HetNets which hinder overall network performance. It is necessary to find optimal resource allocation strategies which can maximize capacity by mitigating interference to harvest the benefits of HetNets. This article formulates the problem of capacity maximization in device-to-device (D2D) assisted heterogeneous relay networks, using mixed integer nonlinear programming (MINLP). MINLP problems are NP-hard in general. Exhaustive search cannot be applied as search space of problem increases exponentially with number of users. Outer approximation algorithm (OAA) is proposed to solve capacity maximization problem. Proposed algorithm is evaluated by extensive simulation work which show that proposed approach is effective in terms of major key performance indicators (KPIs). KPIs considered for algorithm evaluation include user throughput (sum rate) and capacity in terms of number of users accommodated.

Lagrangian relaxation for the reliable shortest path problem with correlated link travel times

Finding a reliable shortest path (RSP) in a stochastic network is a fundamental problem in transportation science. Link travel time correlation significantly affects path reliability, but also greatly increases the complexity of the RSP problem due to the quadratic form of the standard deviation term. Lagrangian relaxation (LR) based on problem reformulation, which only needs to solve a series of shortest path problems, has been recognized as an efficient method to obtain near-optimal RSPs with the optimality gap guarantee. This paper proposes a novel LR approach based on a new convex problem reformulation, and new methods to update Lagrangian multipliers and handle negative cycles of the resulting shortest path problems. Different from existing LR approaches, which adopt the classical subgradient method to solve the dual problem, a constraint generation (CG) algorithm and a subgradient projection (SP) algorithm are proposed to update Lagrangian multipliers effectively, and both algorithms are further modified to handle negative cycles. We also reveal the connection between different reformulations of the RSP problem and show that the proposed approach has a smaller duality gap than existing ones. Experiments on real transportation networks validate the effectiveness of the proposed approach in terms of convergence rate, run time, duality gap and optimality by comparison with the existing LR approaches and the outer approximation algorithm.

Multi-Period Portfolio Optimization with Cone Constraints and Discrete Decisions

In this study, we consider multi-period portfolio optimization model that is formulated as a mixed-integer second-order cone programming problems (MISOCPs). The Markowitz (1952) mean/variance framework has been extended by including transaction costs, conditional value-at-risk (CVaR), diversification-by-sector and buy-in thresholds constraints. The model is obtained using a binary scenario tree that is constructed with monthly returns of the stocks from the S&P 500. We solve these models with a MATLAB based Mixed Integer Linear and Nonlinear Optimizer (MILANO). Numerical results show that we can solve small to medium-sized instances successfully, and we provide a substantial improvement in runtimes using warmstarts in outer approximation algorithm.

On Rate Region of Caching Problems With Non-Uniform File and Cache Sizes

Rate region of the caching problems with non-uniform file and cache sizes can be characterized in terms of the region of entropic vectors, which is unknown in general. One can replace the region of entropic vectors with its polyhedral outer bound to get the associated outer bound on the rate region. A new computation method is introduced for the calculation. The polyhedral projection is converted into a multiple objective linear programming and an implementation of Benson’s outer approximation algorithm is used to solve it. As a demonstration, rate region of the two-file two-user caching problem is calculated and proved to be binary codes sufficient.

Energy-Efficient Cooperative Network Coding With Joint Relay Scheduling and Power Allocation

The energy efficiency (EE) of a multi-user multi-relay system with the maximum diversity network coding (MDNC) is studied. We explicitly find the connection among the outage probability, energy consumption, and EE, and formulate the maximizing EE problem under the outage probability constraint. Relay scheduling (RS) and power allocation (PA) are applied to schedule the relay states (transmitting, sleeping, and so on) and optimize the transmitting power under the practical channel and power consumption models. Since the optimization problem is NP hard, to reduce computational complexity, the outage probability is first tightly approximated to a log-convex form. Furthermore, the EE is converted into a subtractive form based on the fractional programming. Then, a convex mixed-integer nonlinear problem is eventually obtained. With a generalized outer approximation algorithm, RS and PA are solved in an iterative manner. The Pareto-optimal curves between the EE and the target outage probability show the EE gains from PA and RS. Moreover, by comparing with the no network coding (NoNC) scenario, we conclude that with the same number of relays, MDNC can lead to EE gains. However, if RS is implemented, NoNC can outperform MDNC in terms of the EE when more relays are needed in the MDNC scheme.

Outer approximation algorithm with physical domain reduction for computer‐aided molecular and separation process design

Integrated approaches to the design of separation systems based on computer‐aided molecular and process design (CAMPD) can yield an optimal solvent structure and process conditions. The underlying design problem, however, is a challenging mixed integer nonlinear problem, prone to convergence failure as a result of the strong and nonlinear interactions between solvent and process. To facilitate the solution of this problem, a modified outer‐approximation (OA) algorithm is proposed. Tests that remove infeasible regions from both the process and molecular domains are embedded within the OA framework. Four tests are developed to remove subdomains where constraints on phase behavior that are implicit in process models or explicit process (design) constraints are violated. The algorithm is applied to three case studies relating to the separation of methane and carbon dioxide at high pressure. The process model is highly nonlinear, and includes mass and energy balances as well as phase equilibrium relations and physical property models based on a group‐contribution version of the statistical associating fluid theory (SAFT‐γ Mie) and on the GC+ group contribution method for some pure component properties. A fully automated implementation of the proposed approach is found to converge successfully to a local solution in 30 problem instances. The results highlight the extent to which optimal solvent and process conditions are interrelated and dependent on process specifications and constraints. The robustness of the CAMPD algorithm makes it possible to adopt higher‐fidelity nonlinear models in molecular and process design. © 2016 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 62: 3484–3504, 2016

Scheduling of cracking production process with feedstocks and energy constraints

This paper addresses the short-term scheduling problem for the ethylene cracking process with feedstocks and energy constraints. The cracking production of ethylene is a process with units that have decaying performance, requiring periodic cleanup to restore their performance. Under the condition of limited feedstocks, the production operating mode of the cracking furnaces is to keep yields constant by continuously increasing the coil temperature. We present a hybrid MINLP/GDP formulation based on continuous-time representation for the scheduling problem over a finite time horizon. In order to solve the proposed model, which is reformulated as an MINLP model, an improved outer approximation algorithm with multi-generation cuts and problem-dependent integer cuts are developed to solve real large-scale problems. Numerical examples are presented to illustrate the application of the model. Based on analyzing the optimal solution and sensitivity of the model, some conclusions are obtained to provide useful suggestions for real cracking process production.

SCUC‐based optimal power tracing approach for scheduling physical bilateral transactions and its verification via an integrated power‐money flow analysis

This study discusses a security-constrained unit commitment (SCUC) based optimal power tracing approach, which adopts the proportional power tracing method to trace power flows of the network for simultaneously satisfying physical contract paths and financial contract quantities of bilateral transactions. Thus, optimal solutions of the proposed model, including unit commitment and generation dispatch of generators and angle statuses of phase shifters, would simultaneously meet physical and financial requirements of bilateral transactions, and in turn reduce the impacts of loop flows induced by bilateral transactions to third parties of the networked system. The proposed model is a mixed integer non-linear programming problem because of the non-linear proportional power tracing constraints, and the revised outer approximation algorithm is discussed to effectively solve the problem. The effectiveness of the proposed model is further evaluated via an integrated power-money flow analysis, based on the locational marginal price based energy payments and the min–max fairness policy based transmission usage charges. Numerical case studies show that the proposed model, as compared with traditional financial bilateral transaction models, presents potential advantages to avoid loop flows and reduce the impacts to third parties in terms of energy and transmission usage payments.

Process simulator-based optimization of biorefinery downstream processes under the Generalized Disjunctive Programming framework

Downstream processing of biofuels and bio-based chemicals represents a challenging problem for process synthesis and optimization, due to the intrinsic nonideal thermodynamics of the liquid mixtures derived from the (bio) chemical conversion of biomass. In this work, we propose a new interface between the process simulator PRO/II (SimSci, Schneider-Electric) and the optimization environment of GAMS for the structural and parameter optimization of this type of flowsheets with rigorous and detailed models. The optimization problem is formulated within the Generalized Disjunctive Programming (GDP) framework and the solution of the reformulated MINLP problem is approached with a decomposition strategy based on the Outer-Approximation algorithm, where NLP subproblems are solved with the derivative free optimizer belonging to the BzzMath library, and MILP master problems are solved with CPLEX/GAMS. Several validation examples are proposed spanning from the economic optimization of two different distillation columns, the dewatering task of diluted bio-mixtures, up to the distillation sequencing with simultaneous mixed-integer design of each distillation column for a quaternary mixture in the presence of azeotropes.

Integrated Oil-Field Management: From Well Placement and Planning to Production Scheduling

Integrated management can benefit oil-field development and exploitation tremendously. It involves holistic decisions on the order, placement, timing, capacities, and allocations of new well drillings and surface facilities such as manifolds, surface centers, and their interconnections, along with well production/injection profiles. These decisions have profound long-term impacts on field productivity; however, the dynamic nature of oil reservoirs makes them strongly intertwined and highly complex. Hence, a dynamic, holistic, and integrated approach is necessary. Most existing well placement studies ignore surface effects and drilling-rig availability and assume that all wells are opened simultaneously at the beginning of the production horizon. In this work, we extend our previous study [Tavallali Ind. Eng. Chem. Res. 2014, 53 (27), 11033] and develop a mixed integer nonlinear programming (MINLP) approach for addressing such limiting assumptions. We develop a revised outer-approximation algorithm involvi...

Energy Efficient Resource Allocation in D2D-Assisted Heterogeneous Networks with Relays

Heterogeneous networks (HetNets) supported with relays and device to device communication can be considered as a promising solution to realize the ambitious targets of the future fifth generation networks in terms of energy efficiency and capacity. HetNets necessitate optimal power allocation, spectrum resource allocation, and cell selection to meet the quality of service requirements. In this paper, we formulate energy efficiency maximization problem in terms of resource allocation and cell selection for HetNets, where objective is to maximize the network throughput per unit network power consumption. Formulated optimization problem is non-linear fractional programming problem. We use Charnes–Cooper transformation to convert proposed fractional programming problem into concave optimization problem. We propose outer approximation algorithm (OAA) to solve the converted concave optimization problem. The proposed algorithm is evaluated by extensive simulation work. The performance of $\varepsilon $ -optimal solution obtained by OAA method is shown for different network parameters, such as number of users, required date rate, and capacity of network.

Robust beamforming designs for multiuser MISO downlink with per-antenna power constraints

This paper studies the robust beamforming designs for a multiuser multiple-input single-output (MISO) downlink system. Different from the conventional sum-power constraint across all transmit antennas, we consider individual power constraints per antenna at the base station. Assuming that the channel uncertainty is bounded by a spherical region, we develop the optimal robust designs to maximize the minimum worst-case signal-to-interference-plus-noise ratio (SINR) among all users. Specifically, we show that the optimal max-min SINR beamformers can be obtained by solving a sequence of “dual” min-max power problems. Relying on the S-procedure and the linear matrix inequality representation for the cone of Lorentz-positive maps (LPMs), respectively, two designs, which are referred to as Robust-SP and Robust-LPM, are proposed to efficiently solve such min-max power problems. Building on either Robust-SP or Robust LPM, a bisection search algorithm is then developed to find the robust max-min SINR beamformers with guaranteed global optimality and geometrically fast convergence speed. Using the max-min SINR solutions as a cornerstone, we further put forth the optimal robust design for the worst-case weighted sum-rate (WSR) maximization. By formulating the worst-case WSR maximization problem into a monotonic program (MP), we develop a polyblock outer approximation algorithm to obtain the globally optimal solution. Numerical results are presented to demonstrate the merits of the proposed robust beamforming designs.

On speed scaling via integer programming

We consider a class of convex mixed-integer nonlinear programs motivated by speed scaling of heterogeneous parallel processors with sleep states and convex power consumption curves. We show that the problem is NP-hard and identify some polynomially solvable classes. Furthermore, a dynamic programming and a greedy approximation algorithms are proposed to obtain a fully polynomial-time approximation scheme for a special case. For the general case, we implement an outer approximation algorithm.

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson’s outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson’s algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems.

Using multiobjective optimization to map the entropy region

Mapping the structure of the entropy region of at least four jointly distributed random variables is an important open problem. Even partial knowledge about this region has far reaching consequences in other areas in mathematics, like information theory, cryptography, probability theory and combinatorics. Presently, the only known method of exploring the entropy region is, or equivalent to, the one of Zhang and Yeung from 1998. Using some non-trivial properties of the entropy function, their method is transformed to solving high dimensional linear multiobjective optimization problems. Benson’s outer approximation algorithm is a fundamental tool for solving such optimization problems. An improved version of Benson’s algorithm is presented, which requires solving one scalar linear program in each iteration rather than two or three as in previous versions. During the algorithm design, special care is taken for numerical stability. The implemented algorithm is used to verify previous statements about the entropy region, as well as to explore it further. Experimental results demonstrate the viability of the improved Benson’s algorithm for determining the extremal set of medium-sized numerically ill-posed optimization problems. With larger problem sizes, two limitations of Benson’s algorithm is observed: the inefficiency of the scalar LP solver, and the unexpectedly large number of intermediate vertices.