Mixed Integer Quadratic Programming Solver Research Articles

Multidisciplinary Design Optimization Approach to Integrated Space Mission Planning and Spacecraft Design

Space mission planning and spacecraft design are tightly coupled and need to be considered together for optimal performance; however, this integrated optimization problem results in a large-scale mixed-integer nonlinear programming (MINLP) problem, which is challenging to solve. In response to this challenge, this paper proposes a new solution approach to this problem based on decomposition-based optimization via augmented Lagrangian coordination. The proposed approach leverages the unique structure of the problem that enables its decomposition into a set of coupled subproblems of different types: a mixed-integer quadratic programming (MIQP) subproblem for mission planning, and one or more nonlinear programming (NLP) subproblem(s) for spacecraft design. Because specialized MIQP or NLP solvers can be applied to each subproblem, the proposed approach can efficiently solve the otherwise intractable integrated MINLP problem. An automatic and effective method to find an initial solution for this iterative approach is also proposed so that the optimization can be performed without a user-defined initial guess. The demonstration case study shows that, compared to the state-of-the-art method, the proposed formulation converges substantially faster and the converged solution is at least the same or better given the same computational time limit.

FPGA Implementation Framework for Explicit Hybrid Model Predictive Control

Hybrid systems can be controlled by a Hybrid Model Predictive Control (HMPC) with an accurate prediction model and proper formulation. In the real-time evolution of HMPC, an online optimization (mostly, Mixed Integer Quadratic Programming (MIQP)) problem needs to be solved at each sample time. Online implementation of MIQP solver on resource-limited embedded platforms poses challenges, which limits its use for the slow dynamic systems. The online burden of MIQP solver can be moved offline by multi-parametric programming. In this paper, we present an implementation of explicit HMPC on Field Programmable Gate Array (FPGA) for the control of Mixed Logical Dynamical (MLD) systems. We show the step-by-step procedure of FPGA implementation of HMPC considering the model of the inverted pendulum. In the implementation, we used MATLAB-based Hybrid Toolbox to construct hybrid MPC and subsequently generate a low-level C/C++ code of the HMPC. Further, the generated C/C++ code is optimized for memory, speed, and resource utilization by the customized approach of applying pipelining and directives using Xilinx Vivado. The customized explicit HMPC is then implemented on a Xilinx ZYNQ-7000 SoC ZC706 FPGA. The implemented approach is compared with the linear (online and explicit) MPC with a linearized model of the system. The detailed analysis of controller computational complexity in terms of memory, resource utilization, clock, and power consumption is presented. The performance of implemented HMPC is verified through Hardware-in-the-Loop (HIL) co-simulations.

Liquid Air Energy Storage as a polygeneration system to solve the unit commitment and economic dispatch problems in micro-grids applications

Storage technologies play a crucial role in polygeneration plants that attempt to integrate power, thermal and cooling energy systems in order to maximize process efficiency and reduce operating cost. With the increasing penetration of renewable energy into the plant, storage technologies help to dampen the intermittency problem in their energy supply whilst at the same time perform peak shaving to reduce primary energy consumption, thus mitigating pollutant emission. Among the various storage technologies, Liquid Air Energy Storage (LAES) have gathered research interest due to its capability of simultaneously producing electrical and cooling power. Furthermore, unlike Electrochemical Energy Storage (EES) technologies, the LAES lifetime is not heavily dependent on its duty cycle, thus allowing for a calendar life twice or thrice that of EES. In this paper, the economic dispatch of an Eco-building in Singapore has been evaluated using a mixed-integer quadratic programming solver by comparing the adoption of EES and LAES within a capacity range of 300kWh-2000kWh. At the higher end of the capacity range, the LAES configuration results in a higher Net Present Value after 20 years and a shorter time period to obtain the Return of Investment compared to that of EES. At the lower capacity range, both technologies give similar financial returns. Analysis of the results show LAES to be a promising technology to compete with EES in the context of a polygeneration plant and further technology integration is discussed.

Optimal Voltage Regulation of Distribution Networks With Cascaded Voltage Regulators in the Presence of High PV Penetration

This work proposes a novel zone-based multistage “time-graded” operation of cascaded on-load tap changing transformers, capacitor banks, and step voltage regulators in the presence of large-scale photovoltaic (PV) sources. A multistage Volt-VAr optimization (VVO) algorithm is proposed to regulate the voltage in a medium voltage unbalanced distribution system while trying to relax the tap/switch operations of regulators that are cascaded in series, and minimize the curtailment of PV inverter output. A linearized power system model is formulated and a mixed-integer quadratic programming solver is utilized to solve the subobjective optimization at each stage in near to real time. The multistage coordinated operations are performed successively based on the regulator zones starting from the zone nearest to the substation, to achieve the overall voltage regulation of the system. Simulation studies were performed for various scenarios of PV and load profile variations. A comparison study between the novel multistage VVO and a multiobjective VVO was done to observe the convergence performance. Results show the efficient usage of the conventional voltage regulating devices along with minimal power curtailment from PVs when required.

A Numerically Robust Mixed-Integer Quadratic Programming Solver for Embedded Hybrid Model Predictive Control

Abstract The deployment of hybrid model predictive control (MPC) in practical applications requires primarily an efficient and robust on-line Mixed-Integer Quadratic Programming (MIQP) solver that runs in real time. In this paper we propose a new algorithm for solving MIQP problems which is particularly tailored to solve small-scale MIQPs, such as those that arise in embedded hybrid MPC applications. The algorithm couples a branch and bound (B&B) scheme with a recently proposed numerically robust Quadratic Programming (QP) solver based on nonnegative least squares (NNLS) and proximal-point iterations. The resulting MIQP solver supports positive semidefinite Hessian matrices, often appearing in hybrid MPC formulations, and warm starts with respect to both binary and real variables. We show that the speed of execution of our solver is comparable with state-of-the-art commercial solvers, while it is relatively simple to code in an embedded control system.

Review on Reformulation of the Mean-Variance Model with Real-life Trading Restrictions

In this paper, we consider a class of portfolio selection problems with cardinality and minimum buy-in threshold constraints in real-life which can be formulated as mixed-integer quadratic programming (MIQP). Two reformulation methods that generate the same tight continuous relaxation of original problem are compared in the context under the branch-and-bound algorithm, one is the Perspective Reformulation and another is the Lift-and-Convexification Reformulation (LCR). Computational results show that the (PC) is more competitive than the (LCR) method in terms of computing time and nodes in MIQP solver CPLEX 12.7, what's more, this outperformance becomes more obvious as the size of instances grows.

Minimizing the tracking error of cardinality constrained portfolios

We study the problem of selecting a restricted number of shares included in a stock market index, such that the portfolio resembles the index as closely as possible. To measure the difference between the portfolio and the index, referred to as the tracking error, we use a quadratic function with the covariance matrix of the index returns as coefficient matrix. The problem is proved to be strongly NP-hard, and we give theoretical evidence that continuous relaxations of mixed integer quadratic programming (MIQP) formulations are likely to produce poor lower bounds on the tracking error. For fast computation of near-optimal portfolios, we demonstrate how the best-extension-by-one construction heuristic can be designed to run in time bounded by a fourth order polynomial. We also show that the running time of one iteration of the best-exchange-by one improvement heuristic is of the same order. Computational experiments applied to real-life stock market indices show that in instances where an index of less than 500 assets is to be tracked by a portfolio of 10 assets, a commercially available MIQP solver fails to reduce the integrality gap below 94% in 30 CPU-minutes. In contrast, the construction heuristic under study needs less than 30 CPU-seconds to produce a portfolio of 100 assets tracking an index of nearly 2000 assets.

Linear solution schemes for Mean-SemiVariance Project portfolio selection problems: An application in the oil and gas industry

We study the Mean-SemiVariance Project (MSVP) portfolio selection problem, where the objective is to obtain the optimal risk-reward portfolio of non-divisible projects when the risk is measured by the semivariance of the portfolio׳s Net-Present Value (NPV) and the reward is measured by the portfolio׳s expected NPV. Similar to the well-known Mean-Variance portfolio selection problem, when integer variables are present (e.g., due to transaction costs, cardinality constraints, or asset illiquidity), the MSVP problem can be solved using Mixed-Integer Quadratic Programming (MIQP) techniques. However, conventional MIQP solvers may be unable to solve large-scale MSVP problem instances in a reasonable amount of time. In this paper, we propose two linear solution schemes to solve the MSVP problem; that is, the proposed schemes avoid the use of MIQP solvers and only require the use of Mixed-Integer Linear Programming (MILP) techniques. In particular, we show that the solution of a class of real-world MSVP problems, in which project returns are positively correlated, can be accurately approximated by solving a single MILP problem. In general, we show that the MSVP problem can be effectively solved by a sequence of MILP problems, which allow us to solve large-scale MSVP problem instances faster than using MIQP solvers. We illustrate our solution schemes by solving a real MSVP problem arising in a Latin American oil and gas company. Also, we solve instances of the MSVP problem that are constructed using data from the PSPLIB library of project scheduling 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.

Optimizing 4D cone beam computed tomography acquisition by varying the gantry velocity and projection time interval

Four dimensional cone beam computed tomography (4DCBCT) is an emerging clinical image guidance strategy for tumour sites affected by respiratory motion. In current generation 4DCBCT techniques, both the gantry rotation speed and imaging frequency are constant and independent of the patient’s breathing which can lead to projection clustering. We present a mixed integer quadratic programming (MIQP) model for respiratory motion guided-4DCBCT (RMG-4DCBCT) which regulates the gantry velocity and projection time interval, in response to the patient’s respiratory signal, so that a full set of evenly spaced projections can be taken in a number of phase, or displacement, bins during the respiratory cycle. In each respiratory bin, an image can be reconstructed from the projections to give a 4D view of the patient’s anatomy so that the motion of the lungs, and tumour, can be observed during the breathing cycle. A solution to the full MIQP model in a practical amount of time, 10 s, is not possible with the leading commercial MIQP solvers, so a heuristic method is presented. Using parameter settings typically used on current generation 4DCBCT systems (4 min image acquisition, 1200 projections, 10 respiratory bins) and a sinusoidal breathing trace with a 4 s period, we show that the root mean square (RMS) of the angular separation between projections with displacement binning is 2.7° using existing constant gantry speed systems and 0.6° using RMG-4DCBCT. For phase based binning the RMS is 2.7° using constant gantry speed systems and 2.5° using RMG-4DCBCT. The optimization algorithm presented is a critical step on the path to developing a system for RMG-4DCBCT.

A support vector machine with integer parameters

We describe here a method for building a support vector machine (SVM) with integer parameters. Our method is based on a branch-and-bound procedure, derived from modern mixed integer quadratic programming solvers, and is useful for implementing the feed-forward phase of the SVM in fixed–point arithmetic. This allows the implementation of the SVM algorithm on resource–limited hardware like, for example, computing devices used for building sensor networks, where floating–point units are rarely available. The experimental results on well–known benchmarking data sets and a real–world people-detection application show the effectiveness of our approach.

Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem

In this paper, we consider problem (P) of minimizing a quadratic function q(x)=x tQx+ctx of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective function q(x). A classical trick is to raise up the diagonal entries of Q by a vector u until (Q+diag(u)) is positive semidefinite. Then, using the fact that xi2=xi, we can obtain an equivalent convex objective function, which can then be handled by an MIQP solver. Hence, computing a suitable vector u constitutes a preprocessing phase in this exact solution method. We devise two different preprocessing methods. The first one is straightforward and consists in computing the smallest eigenvalue of Q. In the second method, vector u is obtained once a classical SDP relaxation of (P) is solved.We carry out computational tests using the generator of (Pardalos and Rodgers, 1990) and we compare our two solution methods to several other exact solution methods. Furthermore, we report computational results for the max-cut problem.

A Steiner arborescence model for the feeder reconfiguration in electric distribution networks

In this paper we address the problem of finding the radial configuration of an electric distribution network that minimizes the total losses due to the Joule effect. We propose an interpretation of the feeder reconfiguration problem as a Steiner arborescence problem, formulated through a model with a separable quadratic objective function. The problem is then solved by a mixed-integer quadratic programming solver. Computational experience on test networks is reported, showing the effectiveness of the formulation.