Mixed Integer Quadratic Programming Problem Research Articles

Bounds for convex quadratic programming problems and some important applications

This paper proposes bounds for the objective function of convex quadratic programming problem (QPP) in general form. The method uses just the eigenvalues of the Hessian matrix. By use of the eigenvalues of the Hessian matrix and solving two simple optimisation problems, we propose an interval which contains the optimal value of QPP. This method can be useful for complicated and large scale optimisation problems as well as for the integer quadratic programming problems and also can be used as a start interval for the other existing mathematical methods for QPP. Another application of the proposed interval is to help the decision maker in real applications to estimate the bounds of the optimal solution. Thus the method is useful from both theoretical and practical approaches. This method can also be applied to solve fractional quadratic programming problems as well as for binary and mixed integer quadratic programming problems. Sensitivity analysis for the objective function is another application of the method which will be discussed. To illustrate the method, several problems are solved.

A novel projected two-binary-variable formulation for unit commitment in power systems

The thermal unit commitment (UC) problem in power systems can usually be formulated as a mixed-integer quadratic programming (MIQP) problem, which is an NP-hard problem for practical-scale systems and thus is difficult to solve efficiently. In this paper, by projecting the unit generation level onto the interval [0,1] and using reformulation techniques, a novel two-binary-variable (2-bin) MIQP formulation for the UC problem is proposed. The proposed 2-bin formulation is more compact than the state-of-the-art one-binary-variable (1-bin) and three-binary-variable (3-bin) formulations. Moreover, the 2-bin formulation is tighter than the 1-bin and 3-bin formulations in terms of the quadratic cost function, and it is tighter than the 1-bin formulation in terms of linear constraints. The proposed model was tested on 73 instances, including 43 realistic instances and 30 8-unit-based instances, over a scheduling period of 24h for systems ranging from 10 to 1040 generating units. The simulation results show that our proposed MIQP UC formulation is the tightest and most compact model and can be solved most efficiently. After introducing a sequence of piecewise perspective cuts to approximate the quadratic operational cost function, the three UC MIQP formulations can be approximated by three corresponding mixed-integer linear programming (MILP) formulations. Our experiments show that the proposed 2-bin MILP formulation also performs the best in terms of solution times.

POP – Parametric Optimization Toolbox

In this paper, we describe POP, a MATLAB toolbox for parametric optimization. It features (a) efficient implementations of multiparametric programming problem solvers for multiparametric linear and quadratic programming problems and their mixed-integer counter-parts, (b) a versatile problem generator capable of creating random multiparametric programming problems of arbitrary size, and (c) a comprehensive library of multiparametric programming test problems featuring benchmark test sets for multiparametric linear, quadratic, mixed-integer linear, and mixed-integer quadratic programming problems. In addition, POP is equipped with a graphical user interface which enables the user-friendly use of all functionalities of POP and a link to the solvers of the Multi-Parametric Toolbox (MPT), as well as the ability to design explicit MPC problems. These features are demonstrated in detailed computational studies providing insights into the versatility and applicability of POP. Additionally, the example of a period...

Directed graph-based distribution network reconfiguration for operation mode adjustment and service restoration considering distributed generation

We present a directed graph-based method for distribution network reconfiguration considering distributed generation. Two reconfiguration situations are considered: operation mode adjustment with the objective of minimizing active power loss (situation I) and service restoration with the objective of maximizing loads restored (situation II). These two situations are modeled as a mixed integer quadratic programming problem and a mixed integer linear programming problem, respectively. The properties of the distribution network with distributed generation considered are reflected as the structure model and the constraints described by directed graph. More specifically, the concepts of “in-degree” and “out-degree” are presented to ensure the radial structure of the distribution network, and the concepts of “virtual node” and “virtual demand” are developed to ensure the connectivity of charged nodes in every independent power supply area. The validity and effectiveness of the proposed method are verified by test results of an IEEE 33-bus system and a 5-feeder system.

Predictive path following with arrival time awareness for waterborne AGVs

Large ports are seeking innovative logistical ways to improve their competitiveness world-wide. This article proposes waterborne AGVs, inspired by conventional automated guided vehicles and autonomous surface vessels, for transport over water. A predictive path following with arrival time awareness controller is proposed for such waterborne AGVs. The controller is able to achieve smooth tracking and energy efficiency with arrival time awareness for transport oriented applications. Tracking errors are conveniently formulated with vessel dynamics modeled in connected reference path coordinate systems and a coordinate transformation at switching coordinate systems. Binary decision variables and logic constraints based on an along-track state are proposed for modeling switches in the framework of Model Predictive Control (MPC) so that overshoots are avoided. Moreover, timing-aware along-track references are generated by a two-level double integrator scheme. The lower level is embedded in online MPC optimizations for smooth tracking. The higher level solves a mixed-integer quadratic programming problem considering distance-to-go and time-to-go before each MPC optimization. References over the next prediction horizon are generated being aware of the requirements on arrival time. Furthermore, successive linearizations of nonlinear vessel dynamics about a shifted previous optimal system trajectory are implemented to maintain a trade-off between computational complexity and optimality. Simulation results of two industrially relevant Inter Terminal Transport case studies illustrate the effectiveness of the proposed modeling and control design for waterborne AGVs.

A Bi-Level Branch and Bound Method for Economic Dispatch With Disjoint Prohibited Zones Considering Network Losses

This paper proposes a bi-level branch-and-bound ((B&B) method to solve the economic dispatch problem with prohibited zones and network losses. The approach employs binary variables for each prohibited zone and utilizes the B-coefficient for network losses, which can be transformed into a mixed-integer quadratically constrained quadratic programming (MIQCQP), where linear relaxation technique is applied on each bilinear term. Due to the complexity in solving the MIQCQP problem, this paper proposes a bi-level B&B method to achieve global optimum. A spatial B&B method is utilized in the higher level to solve the quadratically constrained quadratic programming (QCQP) problem, whereas a simple B&B method is employed in the lower level to solve a mixed-integer quadratic programming (MIQP) problem. The bi-level B&B algorithm that combines spatial and simple B&B methods is actually a deterministic optimization method and can produce global optimal solutions. Numerical results on 6-unit, 15-unit, and 40-unit test systems show that the bi-level B&B method can solve the MIQCQP problem with superior solution quality and convergence characteristics.

Optimal control of multi‐hop control networks based on the MLD framework

A multi‐hop control network (MHCN) is a control system in which plants and controllers are connected through a multi‐hop wireless network modeled by a directed graph. In this paper, based on the MLD (mixed logical dynamical) framework, which is one of the powerful methods in hybrid systems control, we propose a modeling method and an optimal control method of MHCNs. First, a directed graph in MHCNs and a switch of the control input are modeled by a pair of a linear state equation and a linear inequality with binary variables. Thus, the MLD model expressing a given MHCN is derived. Next, using the MLD model, the optimal control problem is transformed into a mixed integer quadratic programming problem. Finally, numerical examples are presented. The proposed method provides us a basic result for control of MHCNs. © 2015 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

Explicit hybrid model-predictive control: The exact solution

This article presents an algorithm for the exact solution of explicit hybrid model-predictive control problems of time-invariant, discrete-time mixed logical dynamical systems. Using multiparametric programming, this control problem is formulated as a multiparametric mixed-integer quadratic programming problem where the initial states of the system are treated as parameters. In conjunction with decomposition type or branch-and-bound type approaches, the proposed solution strategy first creates affine relaxations of nonconvex critical regions that result from the comparison of two quadratic objective functions. These relaxations are then used to generate an affine outer approximation for the critical regions. Upon termination, the bounded space of the initial states is partitioned into possibly nonconvex critical regions and corresponding optimal control laws.

Distribution System Optimization Based on a Linear Power-Flow Formulation

In this paper, a framework for distribution system optimization is proposed. In this framework, different control variables, such as switchable capacitors, voltage regulators, and system configuration can be optimally determined to satisfy objectives, such as loss minimization and voltage profile improvement. Linearized power-flow equations are used in the optimization, and the problem is formulated as mixed-integer quadratic programming (MIQP), which has a guaranteed optimal solution. Existing efficient solution algorithms developed for MIQP problems facilitate the application of the proposed framework. System operational constraints, such as feeder ampacities, voltage drops, radiality, and the number of switching actions are considered in the model. The performance of the proposed framework is demonstrated using a variety of distribution test systems.

Implementation of Mixed-integer Programming on Embedded System

One of the most widespread modern control strategies is the discrete-time Model Predictive Control (MPC) method which requires the solution of the quadratic programming problem. For systems with binary input variables the quadratic problem is replaced by more challenging Mixed-Integer Quadratic Programming (MIQP) problem. The objective of this work is the implementation of MIQP problem solver in a low power embedded computing platform with limited computational power and limited memory. The MIQP problem is solved using branch-and-bound method and the solution of the relaxed original quadratic problems with equality and inequality constraints solved in the nodes of a binary tree is found with interior-point algorithm. A simulation study of the reserve constrained economic dispatch problem for power generators with prohibited zones is presented. Simulation results show the applicability of the proposed solver for small size MIQP problems.

Solving Mixed-Integer Quadratic Programs via Nonnegative Least Squares

This paper proposes a new algorithm for solving Mixed-Integer Quadratic Programming (MIQP) problems. The algorithm is particularly tailored to solving small-scale MIQPs such as those that arise in embedded hybrid Model Predictive Control (MPC) applications. The approach combines branch and bound (B&B) with nonnegative least squares (NNLS), that are used to solve Quadratic Programming (QP) relaxations. The QP algorithm extends a method recently proposed by the author for solving strictly convex QP's, by (i) handling equality and bilateral inequality constraints, (ii) warm starting, and (iii) exploiting easy-to-compute lower bounds on the optimal cost to reduce the number of QP iterations required to solve the relaxed problems. The proposed MIQP algorithm has a speed of execution that is comparable to state- of-the-art commercial MIQP solvers and is relatively simple to code, as it requires only basic arithmetic operations to solve least-square problems.

Subset selection by Mallows’ [formula omitted]: A mixed integer programming approach

This paper concerns a method of selecting the best subset of explanatory variables for a linear regression model. Employing Mallows’ Cp as a goodness-of-fit measure, we formulate the subset selection problem as a mixed integer quadratic programming problem. Computational results demonstrate that our method provides the best subset of variables in a few seconds when the number of candidate explanatory variables is less than 30. Furthermore, when handling datasets consisting of a large number of samples, it finds better-quality solutions faster than stepwise regression methods do.

Model Predictive Formation Control Using Branch-and-Bound Compatible With Collision Avoidance Problems

This paper presents a model predictive control (MPC) approach for multivehicle formation taking into account collision avoidance and velocity limitation with reduced computational burden. The first part of the paper constructs a formation control law using feedback linearization with MPC in order to reduce the optimal control problem to a mixed-integer quadratic programming problem for a group of unicycles. The second part constructs a new branch-and-bound (B& B) -based algorithm for collision-avoidance problems. Numerical examples and experiments show that the proposed method significantly reduces computation time.

Communication and migration energy aware task mapping for reliable multiprocessor systems

Heterogeneous multiprocessor systems-on-chip (MPSoCs) are emerging as a promising solution in deep sub-micron technology nodes to satisfy design performance and power requirements. However, shrinking transistor geometry and aggressive voltage scaling are negatively impacting the dependability of these MPSoCs by increasing the chances of failures. This paper proposes an offline (design-time) task remapping technique to minimize the communication energy and task migration overhead of an application mapped on a heterogeneous multiprocessor system for all processor fault-scenarios. The proposed technique involves two steps–(1) Communication Energy driven Design Space Exploration (CDSE) to select an initial mapping and (2) Communication energy and Migration overhead aware Task Mapping (CMTM) for different fault-scenarios. The CDSE is formulated as a Mixed Integer Quadratic Programming (MIQP) problem and solved using an energy-gradient based heuristic. The CMTM problem is solved using a fast heuristic with the starting mapping selected using CDSE step. The proposed two steps technique is compared with state-of-the-art approaches through rigorous simulations with synthetic and real application graphs. Results demonstrate that the proposed CDSE reduces design space exploration time by 99% with a maximum variation of 5% from the optimal solution obtained by solving the MIQP problem directly. Further, the proposed CMTM reduces communication energy by an average 35% and migration overhead by an average 20% for all single and double fault-scenarios as compared to the existing fault-tolerant techniques. The CMTM also achieves over 30x reductions in execution time for large problem sizes with a maximum deviation of 15% from the minimum communication energy achievable with the given application on a given architecture. For streaming multimedia applications, the proposed technique delivers 50% higher throughput per unit energy as compared to the existing approaches.

A mixed-integer model predictive control formulation for linear systems

Since their inception in the early 1980s industrial model predictive controllers (MPC) rely on continuous quadratic programming (QP) formulations to derive their optimal solutions. More recent advances in mixed-integer programming (MIP) algorithms show that MIP formulations have the potential of being advantageously applied to the MPC problem. In this paper, we present an MIP formulation that can overcome difficulties faced in the practical implementation of MPCs. In particular, it is possible to set explicit priorities for inputs and outputs, define minimum moves to overcome hysteresis, and deal with digital or integer inputs. The proposed formulation is applied to simulated process systems and the results compared with those achieved by a traditional continuous MPC. The solutions of the resulting mixed-integer quadratic programming (MIQP) problems are derived by a computer implementation of the Outer Approximation method (OA) also developed as part of this work.

A Mixed Integer Quadratic Programming Model for the Low Autocorrelation Binary Sequence Problem

In this paper the low autocorrelation binary sequence problem (LABSP) is modeled as a mixed integer quadratic programming (MIQP) problem and proof of the model’s validity is given. Since the MIQP model is semidefinite, general optimization solvers can be used, and converge in a finite number of iterations. The experimental results show that IQP solvers, based on this MIQP formulation, are capable of optimally solving general/skew-symmetric LABSP instances of up to 30/51 elements in a moderate time. ACM Computing Classification System (1998): G.1.6, I.2.8.

Complete Solutions to Mixed Integer Programming

This paper considers a new canonical duality theory for solving mixed integer quadratic programming problem. It shows that this well-known NP-hard problem can be converted into concave maximization dual problems without duality gap. And the dual problems can be solved, under certain conditions, by polynomial algorithms.

Optimal Photovoltaic Array Reconfiguration to Reduce Partial Shading Losses

Partial shading of a photovoltaic array is the condition under which different modules in the array experience different irradiance levels due to shading. This difference causes mismatch between the modules, leading to undesirable effects such as reduction in generated power and hot spots. The severity of these effects can be considerably reduced by photovoltaic array reconfiguration. This paper proposes a novel mathematical formulation for the optimal reconfiguration of photovoltaic arrays to minimize partial shading losses. The paper formulates the reconfiguration problem as a mixed integer quadratic programming problem and finds the optimal solution using a branch and bound algorithm. The proposed formulation can be used for an equal or nonequal number of modules per row. Moreover, it can be used for fully reconfigurable or partially reconfigurable arrays. The improvement resulting from the reconfiguration with respect to the existing photovoltaic interconnections is demonstrated by extensive simulation results.

Probabilistic-constrained optimal control of a class of stochastic hybrid systems

Stochastic hybrid systems have several applications such as biological systems and communication networks, but it is difficult to consider control of general stochastic hybrid systems. In this paper, a class of discrete-time stochastic hybrid systems, in which only discrete dynamics are stochastic, is considered. For this system, a solution method for the optimal control problem with probabilistic constraints is proposed. Probabilistic constraints guarantee that the probability that the continuous state reaches a given unsafe region is less than a given constant. In the propose method, first, continuous state regions, from which the state reaches a given unsafe region, are computed by a backward-reachability graph. Next, mixed integer quadratic programming problems with constraints derived from the backward-reachability graph are solved. The proposed method can be applied to model predictive control.

Biofuel supply chain design under competitive agricultural land use and feedstock market equilibrium

The rapid expansion of the biofuel industry diverts a large amount of agricultural crops as energy feedstocks, and in turn affects farm land allocation, feedstock market equilibrium, and agricultural economic development in local areas. In this paper, we propose game-theoretic models that incorporate farmers' decisions on land use and market choice into the biofuel manufacturers' supply chain design problem. A noncooperative bi-level Stackelberg leader–follower game model and a cooperative game model are developed respectively to address possible business partnership scenarios between feedstock suppliers and biofuel manufacturers. The models determine the optimal number and locations of biorefineries, the required prices for these refineries to compete for feedstock resources, as well as farmers' land use choices between food and energy. Using corn as an example of feedstock crops, spatial market equilibrium is utilized to model the relationship between corn supply and demand, and the associated price variations in local grain markets. With linear corn demand functions, we develop a solution approach that transforms the original discrete mathematical program with equilibrium constraints (DC-MPEC) into to a solvable mixed integer quadratic programming (MIQP) problem based on Karush–Kuhn–Tucker (KKT) conditions. The proposed methodology is illustrated using an empirical case study of the Illinois State. The computation results reveal interesting insights into optimal land use strategies and supply chain design for the emerging “biofuel economy”.