An Application of Interior Point Method to 0-1 Mixed Integer Programing Problems

Future wireless communication systems are expected to provide a broad range of multimedia services that have a significant traffic asymmetry between uplink and downlink. The code division multiple access‐time division duplex (CDMA‐TDD) system is a promising solution to cope with the problem of traffic asymmetry. However, the TDD system is subject to inter‐cell interference compared to frequency division duplex (FDD) system. Since both uplink and downlink share the same frequency in TDD, uplink and downlink may interfere each other especially when neighboring cells require different rates of traffic load. Thus, the resource allocation among cells is an important issue in TDD. In this paper, the resource allocation in the CDMA‐TDD is formulated as a mixed integer programming (MIP) problem. A dynamic resource allocation algorithm (DRAA) is provided that effectively solves the traffic asymmetry problem. The MIP problem is also solved by a well‐known branch and bound procedure. Both the crossed slot and non‐crossed slot allocation are examined and compared to the DRAA. Computational result shows that proposed DRAA gives a good performance as the traffic asymmetry increases between the uplink and downlink. Copyright © 2003 John Wiley & Sons, Ltd.

Real-time power optimization for application server clusters based on Mixed-Integer Programming

Benders’ partitioning method is a classical method for solving mixed integer programming problems. The basic idea is to partition the problem by dividing the variables into complicating variables, normally the variables that are constrained to be integer valued, and easy variables, normally the continuous variables. By fixing the complicating variables, a convex sub-problem is generated. Solving this convex sub-problem and its dual generates cutting planes that are used to create a master problem, which when solved generates new values for the complicating variables, which have a potential to give a better solution. In this work, we assume that the optimal solution is given, and we present a way in which the partitioning idea can be used to generate a valid inequality that supports the optimal solution. By adding some of the continuous variables to the complicating variables, we generate a valid inequality, which is a supporting hyperplane to the convex hull of the mixed integer program. The relaxed original programming problem with this supporting valid inequality added will produce interpretable prices for the original mixed integer programming problem. The method developed can be used to generate economically interpretable prices for markets with non-convexities. This is an important issue in many of the deregulated electricity markets in which the non-convexities comes from large start up costs or block bids. The generated prices are, in the case when the sub-problem generated by fixing the integer variables to their optimal values has the integrality property, also supported by nonlinear price functions that are the basis for integer programming duality.

In this paper, we consider simultaneous wireless information and energy transfer in an orthogonal-frequency-division-multiple-access decode-and-forward relay network, in which an energy-constrained relay node harvests energy from a source node and uses the harvested energy to forward information to multiple destination nodes. Our objective is to maximize the end-to-end sum rate by resource allocation, subject to transmit power constraint at the source and energy-harvesting (EH) constraint at the relay. A non-convex and mixed-integer programming (MIP) problem is formulated to optimize time-switching (TS) ratios of EH and information decoding at the relay, TS ratio of information transmission from relay to destinations, subcarrier allocation as well as power allocation (PA) over all subcarriers at source and relay. We propose to decouple this problem into a convex problem and an MIP problem in fractional form. To solve the MIP problem, we transform it into an equivalent optimization problem in subtractive form which has a tractable solution. As a result, we propose a novel scheme to achieve jointly optimal TS ratios, subcarrier allocation and PA. Simulation results verify the optimality of our proposed resource allocation scheme.

Genetic algorithm for non-linear mixed integer programming problems and its applications

An optimization procedure is described that can serve to identify attractive regionalization plans for large wastewater treatment and collection systems. Potential expansion of facilities in Western Suffolk County, Long Island, New York is considered. A mixed integer programming approach is utilized. Each system element cost, taken as a function of wastewater flow, is represented by a linear variable cost plus a fixed cost. The objective is to minimize total cost while ensuring that all wastewater flow generated is treated. Political and administrative feasibility and the location of effluent discharge constrain the model. The Long Island problem is too large to be solved with a standard formulation of wastewater network as a mixed integer programming problem. To improve computational efficiency, additional constraints are introduced on the minimum flow in each pipe and on the number of pipes leaving each node. Network configurations obtained for Western Suffolk County show that limiting development plans to 201 regions is not cost effective. Regional solutions indicate a general trade‐off between lower costs for coastal discharge and increased groundwater recharge.

Sparse support recovery from multiple measurement vectors (MMV) is studied in this paper. For years, a noticeable mismatch exists between the theories and algorithms in the research of super-resolution and direction-of-arrival (DOA) estimation that variants of separation conditions are assumed to ensure stable recovery while the corresponding algorithms rarely exploit such structural constraints. Due to the discrete nature of the separation condition, we propose to incorporate such prior information in a Mixed Integer Programming (MIP) problem with an <inline-formula><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula>-based constraint. We develop a specialized branch and bound (B&amp;B) algorithm that can efficiently exploit the separation prior with guaranteed complexity reduction. Moreover, we show that computational complexity can be further reduced by leveraging the sparse array idea along with a particular perspective formulation of the MIP. The superior performance of the proposed algorithm is demonstrated via numerical experiments.

A note on a decision support system for multiobjective integer and mixed-integer programming problems

Centralized algorithms are widely used for optimization and control in power system applications. These algorithms require all the measurements and data to be accumulated at central location and hence suffer from single-point-of-failure. Additionally, these algorithms lack scalability with an increasing number of sensors and actuators, specially with the increasing integration of distributed energy resources (DERs). As the power system becomes a confluence of a diverse set of decision-making entities with a multitude of objectives, the preservation of privacy and operation of the system with limited information has been a growing concern. Distributed optimization techniques solve these challenges while also ensuring resilient computational solutions for the power system operation in the presence of both natural and man-made adversaries. A detailed discussion of possible applications of distributed optimization in power systems is provided in this work. However, there exist multiple challenges for accurate and computationally efficient distributed solutions. Commonly-used distributed optimization approaches include Lagrange relaxation, augmented Lagrangian, approximate network directions in conjunction with standard Lagrangian, auxiliary problem principle, alternating directions method of multipliers, optimality condition decomposition, proximal atomic coordination, and optimal feedback-based voltage control. A comprehensive classification of the distributed optimization problems has been discussed and detailed in this work. All of these algorithms have displayed efficient identification of global optimum solutions for convex continuous distributed optimization problems. The algorithms discussed so far are predominantly used to manage continuous state variables. Inclusion of integer variables in the decision support are needed for specific power system problems. Mixed integer programming (MIP) problem arises in a power system operation and control due to tap changing transformers, capacitors and switches. The global optimization techniques for MIPs are Branch and Bound, Branch and Cut, Cutting planes, Adaptive coordinate search, Nelder-Mead, Genetic algorithm etc. Although the above optimization techniques are able to solve NP-hard convexified MIP problems centrally, but are time consuming and do not scale well for large scale distributed problems. Decomposition and solution approach of distributed coordination can resolve the scalability issue. Despite the fact that a large body of work is present on the centralized solution methods for convexified MIP problems, the literature on distributed MIPs is relatively limited. The distributed optimization algorithms applied in power network to solve MIPs are reported here. ML based solutions can help to get faster convergence for distributed optimization or can replace optimization techniques depending on the problem as discussed in this work. Finally, a summary and path forward are provided, and the advancement needed in distributed optimization for the power grid is also presented.

Most real-world optimization problems are multiobjective by nature, involving noncomparable objectives. Many of these problems can be formulated in terms of a set of linear objective functions that should be simultaneously optimized over a class of linear constraints. Often there is the complicating factor that some of the variables are required to be integral. The resulting class of problems is named multiobjective mixed integer programming (MOMIP) problems. Solving these kinds of optimization problems exactly requires a method that can generate the whole set of nondominated points (the Pareto-optimal front). In this paper, we first give a survey of the newly developed branch and bound methods for solving MOMIP problems. After that, we propose a new branch and bound method for solving a subclass of MOMIP problems, where only two objectives are allowed, the integer variables are binary, and one of the two objectives has only integer variables. The proposed method is able to find the full set of nondominated points. It is tested on a large number of problem instances, from six different classes of MOMIP problems. The results reveal that the developed biobjective branch and bound method performs better on five of the six test problems, compared with a generic two-phase method. At this time, the two-phase method is the most preferred exact method for solving MOMIP problems with two criteria and binary variables. This paper was accepted by Dimitris Bertsimas, optimization.

Video delivery networks (VDNs) using cloud storage have recently started to emerge. Different to the conventional VDNs, Cloud Video Delivery Networks (CVDNs) take advantage of the cloud computing technology which helps to reduce the storage and bandwidth cost. However, the existing CVDNs model cannot guarantee the Quality of Service (QoS) adequately because of the dynamic demands of end users such as the delay and the priority. In this paper, we put forward a path-splitting-supported and priority-oriented CVDNs method to increase the acceptance ratio of requests and provide the QoS of end users. In the proposed method, we divide the end users into ordinary users and paying users while the paying users having higher priorities. Then, we formulate the priority-oriented cloud video delivery problem as a mixed integer programming (MIP) problem while considering the routing constraints. The experimental results show that the acceptance rate of path-splitting-supported method is 20% higher than the method of path-unsplitting. In addition, our method can also make CVDNs providers get higher earnings.

Learning k-maxitive fuzzy measures from data by mixed integer programming

As more renewable energy is integrated into the power grid, it is increasingly important to exploit variable electricity pricing structures to minimize commercial utility costs and enable more intermittent renewables on the grid through proactive management of energy storage. Using data from a large campus district energy system, equipped with centralized chilled water plants and a thermal energy storage tank, a novel technique is proposed to optimize this system in real-time, formulated as a mixed-integer quadratic programming problem. This method, titled Quadratic Programming Hybrid with Augmented Constraints, is sufficiently fast to be computed in real-time for a district chiller system. This method is compared to both Branch and Bound and a simple logical decision algorithm in both speed and optimality. The proposed method for solving this mixed-integer quadratic programming problem proves very successful at achieving a near-optimal solution when compared to a standard Branch and Bound (BnB) algorithm. Although suboptimal, the proposed algorithm takes 99.96–99.99% less computational time than the standard BnB and computes an answer to within 29.9% of the BnB objective function. When compared to a simple logical decision algorithm, which represents an operator manually controlling the plant, the proposed method is estimated to yield 8.10–33.7% in savings on chiller energy costs. The Quadratic Programming Hybrid with Augmented Constraints algorithm shows potential for use in a real-time optimization application to exploit variable electricity pricing and significantly reduce the costs of running a chiller plant with thermal energy storage.

Multi-parametric programming based algorithms for the global solution of bi-level mixed-integer linear and quadratic programming problems

Network virtualization, as a fundamental technology of Future Internet Architecture, has been used to overcome the ossification of the current Internet. This paper focuses on virtual nodes splitting for aggregating the diverse physical resources in the issue of virtual network embedding (VNE), which is used for the resource allocation in network virtualization. A mixed integer programming (MIP) problem has been proposed by taking both path-splitting and node-splitting into considerations. The feasibility of virtual node splitting is analyzed and a corresponding formal expression is formulated. The VNE performance is evaluated using by a generic MIP solver over a NSFNET topology. The numerical evaluations in this paper show that, comparing with the approaches for VNE without node-splitting, the proposed algorithm is beneficial to reduce VNE cost, as well as to increase VN acceptance ratio.