A Partitioning Method that Generates Interpretable Prices for Integer Programming Problems

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

Semi-supervised support vector machines arise in machine learning as a model of mixed integer programming problem for classification. In this paper, we propose two convex conic relaxations for the original mixed integer programming problem. The first one is a new semi-definite relaxation, and its possibly maximal ratio of the optimal value is estimated approximately. The second one is a doubly nonnegative relaxation, which is relaxed from a well-known conic programming problem called completely positive programming problem that is equivalent to the original problem. Furthermore, we prove that the doubly nonnegative relaxation is tighter than the semi-definite relaxation. Finally, the numerical results show that two proposed relaxations not only generate proper classifiers but also outperform some existing methods in classification accuracy.

Given an initial marking and final marking for a Petri net model, an optimal firing sequence problem is defined as the problem to find an optimal transition firing sequence to minimize the objective function. For the purpose of analysis of general integer programming problems, we propose a Petri net representation and reachability analysis of integer programming problems. In the proposed method, an integer programming problem is converted into the optimal firing sequence problem of Petri nets. By utilizing the proposed algorithm, integer programming problems can be visualized and analyzed by the Petri net theory. We apply the proposed methodology to the scheduling problems of dual armed cluster tools. The valid inequalities are derived from the reachability analysis. Numerical results show that those valid inequalities can significantly reduce the computational time of the original integer programming problem.

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

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.

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

A partitioning algorithm for the network loading problem

We study the polyhedral structure of simple mixed integer sets that generalize the two variable set {(s,z)R 1 + ×ℤ1:s≥z−b}. These sets form basic building blocks that can be used to derive tight formulations for more complicated mixed integer programs. For four such sets we give a complete description by valid inequalities and/or an integral extended formulation, and we also indicate what constraints can be added without destroying integrality. We apply these results to provide tight formulations for certain piecewise–linear convex objective integer programs, and in a companion paper we exploit them to provide polyhedral descriptions and computationally effective mixed integer programming formulations for discrete lot-sizing 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.

We present a method based on mean field theory to cope with the mixed nonlinear integer programming, especially with optimal power flow problems involving both continuous and discrete variables, in a more exact manner. That is, we first formulate OPF as a mixed integer programming, and then derive its mean field equations as well as the annealing algorithm, by taking advantage of the characteristics of the original problems. Numerical simulations have verified effectiveness of this approach for small power systems.

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.

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

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

We consider the “harvest-then-transmit” protocol in a wireless powered communication network (WPCN), where an energy-constrained access point (AP) harvests energy from the radio-frequency signals transmitted by a power beacon (PB) for assisting user data transmission. In the wireless information transfer (WIT) phase, AP employs the harvested energy to convey independent signals to multiple users through either time-division multiple access (TDMA) or orthogonal frequency-division multiple access (OFDMA). Aiming to maximize the sum rate (SR) of the WPCN, we jointly optimize the energy harvesting (EH) time and the AP power allocation, considering both the conventional linear and practical nonlinear EH models at the AP. The optimization problems of both TDMA- and OFDMA-enabled WPCNs are formulated as nonconvex programs, which are challenging to solve globally. To achieve an efficient optimal solution to the problem of TDMA-enabled WPCN, we first decompose the original nonconvex problem into three convex subproblems, and then propose a low-complexity iterative algorithm for its solution. For the OFDMA-enabled WPCN, the problem belongs to a difficult class of mixed-integer nonconvex programming due to the involvement of binary variables for subcarrier allocation. To overcome this issue, we convert the problem to a quasi-convex problem and then employ a bisection search to obtain the optimal solution. Simulation results are provided to confirm the benefit of jointly optimizing the EH time and the AP power allocation compared to baseline schemes. The performance of the proposed TDMA-enabled WPCN is shown to be superior to that of the proposed OFDMA-enabled WPCN in terms of SR when the transmit power of PB and the number of antennas of AP are relatively large.