Quadratically Constrained Quadratic Programming Research Articles

ANTIGONE: Algorithms for coNTinuous / Integer Global Optimization of Nonlinear Equations

This manuscript introduces ANTIGONE, Algorithms for coNTinuous/Integer Global Optimization of Nonlinear Equations, a general mixed-integer nonlinear global optimization framework. ANTIGONE is the evolution of the Global Mixed-Integer Quadratic Optimizer, GloMIQO, to general nonconvex terms. The purpose of this paper is to show how the extensible structure of ANTIGONE realizes our previously-proposed mixed-integer quadratically-constrained quadratic program and mixed-integer signomial optimization computational frameworks. To demonstrate the capacity of ANTIGONE, this paper presents computational results on a test suite of $$2{,}571$$ 2 , 571 problems from standard libraries and the open literature; we compare ANTIGONE to other state-of-the-art global optimization solvers.

SDR-Based Precoding for Multi-User Multi-Stream MIMO Downlinks

The existing methods based on convex-optimization theory, which use the concept of SINR, can just design the optimal precoder for each user with single antenna. In this paper, we design the optimal precoding matrices for multi-user MIMO downlinks by solving the optimization problem that minimizes total transmit power subject to signal-leakage-plus-noise-ratio (SLNR) constraints. Because SLNR of each user is determined by its own precoding matrix and is independent of other users, the goal problem can be separated into a series of decoupled low-complexity quadratically constrained quadratic programs (QCQPs). Using the semidefinite relaxation (SDR) technique, these QCQPs can be reformulated into the semidefinite programs (SDP) and be solved effectively. Simulation results show that proposed precoding scheme is quite feasible when each user has two receive antennas, and it has better bit error rate (BER) performance than the original maximal-SLNR precoding scheme when SLNR of each user satisfies large threshold value.

Randomized Algorithms for Optimal Solutions of Double-Sided QCQP With Applications in Signal Processing

Quadratically constrained quadratic programming (QCQP) with double-sided constraints has plenty of applications in signal processing as have been addressed in recent years. QCQP problems are hard to solve, in general, and they are typically approached by solving a semidefinite programming (SDP) relaxation followed by a postprocessing procedure. Existing postprocessing schemes include Gaussian randomization to generate an approximate solution, rank reduction procedure (the so-called purification), and some specific rank-one matrix decomposition techniques to yield a globally optimal solution. In this paper, we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double-sided QCQP (i.e., instances with a small number of constraints). We illustrate their applicability in robust receive beamforming, radar optimal code design, and broadcast beamforming for multiuser communications. As a byproduct, we derive an alternative (shorter) proof for the Sturm-Zhang rank-one matrix decomposition theorem.

Optimization of radar phase-coded signals for multiple target detection

We address the design of radar phase-coded signals for the detection of multiple targets in the presence of a coloured Gaussian disturbance with a similarity constraint and a detection constraint. The weighted average signal-to-noise ratio (SNR), which characterises the overall level of multiple targets detection performance, is used as the cost function for the optimization of the radar waveform with a constraint on the degree of similarity with a prefixed radar signal and a constraint on the minimal SNR threshold for every target. We formulate the optimization problem in terms of a non-convex quadratically constrained quadratic program (QCQP) that is NP-hard. Hence, we propose a similarity and detection constrained signal (SDCS) design algorithm relying on both the semidefinite relaxation (SDR) technique and the randomization technique to obtain an accurate approximation of the optimal solution for the case with a continuous phase alphabet and the case with a finite phase alphabet. Finally, the results of extensive simulations demonstrate that the phase-coded signals generated with our approach are accurate approximations of the optimal signals and that a trade-off exists between the detection performance and the requirements on the signal similarity and the minimal SNR threshold.

Fixed-flowrate total water network synthesis under uncertainty with risk management

Abstract This work addresses the problem of integrated water network synthesis under uncertainty with risk management. We consider a superstructure consisting of water sources, regenerators, and sinks that leads to a mixed-integer quadratically-constrained quadratic program (MIQCQP) for a fixed-flowrate total water network synthesis problem. Uncertainty in the problem is accounted for via a recourse-based two-stage stochastic programming formulation with discrete scenarios that gives rise to a multiscenario MIQCQP comprising network design in the first stage and its operation in the second stage acting as recourse. In addition, we extend the model to address risk management using the Conditional Value-at-Risk (CVaR) metric. Because a large number of scenarios is often required to capture the underlying uncertainty of the problem, causing the model to suffer from the curse of dimensionality, we propose a stepwise solution strategy to reduce the computational load. We illustrate this methodology on a case study inspired from the water network of a petroleum refinery in Malaysia. The presence of nonconvex bilinear terms necessitates the use of global optimization techniques for which we employ a new global MIQCQP solver, GAMS/GloMIQO and verify the solutions with BARON. Our computational results show that total water network synthesis under uncertainty with risk management problems can be solved to global optimality in reasonable time.

A Novel Optimization Method for Nonconvex Quadratically Constrained Quadratic Programs

This paper presents a novel optimization method for effectively solving nonconvex quadratically constrained quadratic programs (NQCQP) problem. By applying a novel parametric linearizing approach, the initial NQCQP problem and its subproblems can be transformed into a sequence of parametric linear programs relaxation problems. To enhance the computational efficiency of the presented algorithm, a cutting down approach is combined in the branch and bound algorithm. By computing a series of parametric linear programs problems, the presented algorithm converges to the global optimum point of the NQCQP problem. At last, numerical experiments demonstrate the performance and computational superiority of the presented algorithm.

A branch-and-cut approach to portfolio selection with marginal risk control in a linear conic programming framework

Marginal risk represents the risk contribution of an individual asset to the risk of the entire portfolio. In this paper, we investigate the portfolio selection problem with direct marginal risk control in a linear conic programming framework. The optimization model involved is a nonconvex quadratically constrained quadratic programming (QCQP) problem. We first transform the QCQP problem into a linear conic programming problem, and then approximate the problem by semidefinite programming (SDP) relaxation problems over some subrectangles. In order to improve the lower bounds obtained from the SDP relaxation problems, linear and quadratic polar cuts are introduced for designing a branch-and-cut algorithm, that may yield an ∈-optimal global solution (with respect to feasibility and optimality) in a finite number of iterations. By exploring the special structure of the SDP relaxation problems, an adaptive branch-and-cut rule is employed to speed up the computation. The proposed algorithm is tested and compared with a known method in the literature for portfolio selection problems with hundreds of assets and tens of marginal risk control constraints.

Faster, but weaker, relaxations for quadratically constrained quadratic programs

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.

Design of phase codes with interval staggering for Doppler ambiguous target detection

The design of radar coherent pulse trains encoded with interval staggering for ambiguous Doppler target detection in clutter is addressed. To alleviate the Doppler ambiguous problem, an interpulse phase coding optimisation model is formulated in terms of staggered interval sampling, while maximising the output signal-tointerference- plus-noise ratio for improved target detection. Based on a reformulation of this optimisation problem as a non-convex quadratically constrained quadratic program (QCQP), a semidefinite relaxation algorithm for this QCQP is presented. Numerical results demonstrate the effectiveness of this proposed approach.

Dynamic Economic Dispatch Considering Transmission Losses Using Quadratically Constrained Quadratic Program Method

In response to the challenges brought on by the energy conservation based generation dispatch (ECGD) in China, a dynamic economic dispatch model that considers transmission losses (DED-TL) and a new solution framework are proposed to improve the accuracy, efficiency, and robustness of generation scheduling. Transmission losses are presented explicitly in the model so that the total energy consumption, consisting of both the generators' operation costs and also the transmission losses, can be well considered. The model is formulated as a quadratically constrained quadratic program (QCQP) problem that can be solved by commercial solvers. A penalty-based algorithm is proposed to address the virtual transmission loss problem brought on by negative nodal marginal costs. Three cases-a 6-bus system, the IEEE 30-bus system, and a 2746-bus system based on the electric energy system of Poland-are employed in numerical experiments to demonstrate the effectiveness of the proposed method. The results show that the better scheduling results in higher accuracy, and that good computational efficiency can be acquired via the proposed method.

Using quadratic convex reformulation to tighten the convex relaxation of a quadratic program with complementarity constraints

Quadratic Convex Reformulation (QCR) is a technique that has been proposed for binary and mixed integer quadratic programs. In this paper, we extend the QCR method to convex quadratic programs with linear complementarity constraints (QPCCs). Due to the complementarity relationship between the nonnegative variables $$y$$ and $$w$$ , a term $$y^{T}Dw$$ can be added to the QPCC objective function, where $$D$$ is a nonnegative diagonal matrix chosen to maintain the convexity of the objective function and the global resolution of the QPCC. Following the QCR method, the products of linear equality constraints can also be used to perturb the QPCC objective function, with the goal that the new QP relaxation provides a tighter lower bound. By solving a semidefinite program, an equivalent QPCC can be obtained whose QP relaxation is as tight as possible. In addition, we extend the QCR to a general quadratically constrained quadratic program (QCQP), of which the QPCC is a special example. Computational tests on QPCCs are presented.

Exact Computable Representation of Some Second-Order Cone Constrained Quadratic Programming Problems

Solving the quadratically constrained quadratic programming (QCQP) problem is in general NP-hard. Only a few subclasses of the QCQP problem are known to be polynomial-time solvable. Recently, the QCQP problem with a nonconvex quadratic objective function over one ball and two parallel linear constraints is proven to have an exact computable representation, which reformulates the original problem as a linear semidefinite program with additional linear and second-order cone constraints. In this paper, we provide exact computable representations for some more subclasses of the QCQP problem, in particular, the subclass with one second-order cone constraint and two special linear constraints.

A Heuristic algorithm to solve the unit commitment problem for real-life large-scale power systems

One of the main needs that power system operators around the world have is to solve complex Unit Commitment models for large-scale power systems in an acceptable computation time. This Paper presents an alternative Heuristic algorithm that successfully addresses this need. The Heuristic algorithm makes use of various optimization techniques such as Mixed Integer Linear Programming (MILP), Quadratic Programming (QP), Quadratically Constrained Programming (QCP), and Dynamic Programming (DP). CPLEX 12.2 is used as the main optimization engine for MILP, QP, and QCP. DP is an in-house algorithm used to obtain the commitment of Combined Cycle Plants (CCPs) when represented with the component-based model. This Heuristic algorithm combines the global optimality capabilities of MI (L) P formulations with the highly detailed models available for CCPs using LR–DP formulations. The Heuristic algorithm introduced in this Paper is capable of solving up to 1-week scenarios with a 1-hour time window for the complex Mexican Power System.

A nonlinear norm-relaxed method for finely discretized semi-infinite optimization problems

In this paper, we present a sequential simple quadratically constrained quadratic programming (QCQP) norm-relaxed method for finely discretized semi-infinite optimization problems. At each iteration, the iteration point is feasible, and an improved search direction is solved by only one simple QCQP subproblem, in which only a few of constraints are chosen. Under some weak conditions, the proposed algorithm possesses weak global convergence. Finally, numerical results show that the proposed method is effective.

Fluence map optimization (FMO) with dose–volume constraints in IMRT using the geometric distance sorting method

A new heuristic algorithm based on the so-called geometric distance sorting technique is proposed for solving the fluence map optimization with dose–volume constraints which is one of the most essential tasks for inverse planning in IMRT. The framework of the proposed method is basically an iterative process which begins with a simple linear constrained quadratic optimization model without considering any dose–volume constraints, and then the dose constraints for the voxels violating the dose–volume constraints are gradually added into the quadratic optimization model step by step until all the dose–volume constraints are satisfied. In each iteration step, an interior point method is adopted to solve each new linear constrained quadratic programming. For choosing the proper candidate voxels for the current dose constraint adding, a so-called geometric distance defined in the transformed standard quadratic form of the fluence map optimization model was used to guide the selection of the voxels. The new geometric distance sorting technique can mostly reduce the unexpected increase of the objective function value caused inevitably by the constraint adding. It can be regarded as an upgrading to the traditional dose sorting technique. The geometry explanation for the proposed method is also given and a proposition is proved to support our heuristic idea. In addition, a smart constraint adding/deleting strategy is designed to ensure a stable iteration convergence. The new algorithm is tested on four cases including head–neck, a prostate, a lung and an oropharyngeal, and compared with the algorithm based on the traditional dose sorting technique. Experimental results showed that the proposed method is more suitable for guiding the selection of new constraints than the traditional dose sorting method, especially for the cases whose target regions are in non-convex shapes. It is a more efficient optimization technique to some extent for choosing constraints than the dose sorting method. By integrating a smart constraint adding/deleting scheme within the iteration framework, the new technique builds up an improved algorithm for solving the fluence map optimization with dose–volume constraints.

Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations

We propose a deterministic global optimization approach, whose novel contributions are rooted in the edge-concave and piecewise-linear underestimators, to address nonconvex mixed-integer quadratically-constrained quadratic programs (MIQCQP) to \({\epsilon}\) -global optimality. The facets of low-dimensional (n ≤ 3) edge-concave aggregations dominating the termwise relaxation of MIQCQP are introduced at every node of a branch-and-bound tree. Concave multivariable terms and sparsely distributed bilinear terms that do not participate in connected edge-concave aggregations are addressed through piecewise-linear relaxations. Extensive computational studies are presented for point packing problems, standard and generalized pooling problems, and examples from GLOBALLib (Meeraus, Globallib. http://www.gamsworld.org/global/globallib.htm).

GloMIQO: Global mixed-integer quadratic optimizer

This paper introduces the global mixed-integer quadratic optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs to $${\varepsilon}$$ -global optimality. The algorithmic components are presented for: reformulating user input, detecting special structure including convexity and edge-concavity, generating tight convex relaxations, partitioning the search space, bounding the variables, and finding good feasible solutions. To demonstrate the capacity of GloMIQO, we extensively tested its performance on a test suite of 399 problems of diverse size and structure. The test cases are taken from process networks applications, computational geometry problems, GLOBALLib, MINLPLib, and the Bonmin test set. We compare the performance of GloMIQO with respect to four state-of-the-art global optimization solvers: BARON 10.1.2, Couenne 0.4, LindoGLOBAL 6.1.1.588, and SCIP 2.1.0.

Representing quadratically constrained quadratic programs as generalized copositive programs

We show that any (nonconvex) quadratically constrained quadratic program (QCQP) can be represented as a generalized copositive program. In fact, we provide two representations: one based on the concept of completely positive (CP) matrices over second-order cones, and one based on CP matrices over the positive semidefinite cone.

Robust Multicast Beamforming for Spectrum Sharing-Based Cognitive Radios

We consider a robust downlink beamforming optimization problem for secondary multicast transmission in a multiple-input multiple-output (MIMO) spectrum sharing cognitive radio (CR) network. The minimization of transmit power is formulated subject to both quality-of-service (QoS) constraints on the secondary receivers and interference temperature constraints on the primary users, under the assumption of imperfect channel state information (CSI). The problem is a nonconvex quadratically constrained quadratic program (QCQP), and in general it is hard to achieve the global optimality. As a compromise, we present two randomized approximation algorithms for the problem via convex optimization techniques. Apart from the general setting of the robust beamforming problem, we identify one interesting special case, the robust problem of which can be solved efficiently. Simulation results are presented to demonstrate the performance gains of the proposed algorithms over an existing robust design.

Optimal link removal for epidemic mitigation: A two-way partitioning approach

The structure of the contact network through which a disease spreads may influence the optimal use of resources for epidemic control. In this work, we explore how to minimize the spread of infection via quarantining with limited resources. In particular, we examine which links should be removed from the contact network, given a constraint on the number of removable links, such that the number of nodes which are no longer at risk for infection is maximized. We show how this problem can be posed as a non-convex quadratically constrained quadratic program (QCQP), and we use this formulation to derive a link removal algorithm. The performance of our QCQP-based algorithm is validated on small Erdős–Renyi and small-world random graphs, and then tested on larger, more realistic networks, including a real-world network of injection drug use. We show that our approach achieves near optimal performance and out-performs other intuitive link removal algorithms, such as removing links in order of edge centrality.