Mixed-integer Quadratically-constrained Quadratic Program Research Articles

QoE-Oriented Mobile Virtual Reality Game in Distributed Edge Networks

Mobile edge computing is a promising framework for mobile virtual reality (VR) game. Although there are several existing studies on the edge assisted mobile VR game system, they lack the consideration of provisioning services with satisfactory QoE to a large number of users. In this paper, we consider the problem of providing QoE-oriented edge assisted mobile VR game as a service to multiple users, with a comprehensive QoE concern of both visual and delay aspects. Due to the unique features of mobile VR game, the problem is formulated into a Mixed Integer Quadratically Constrained Quadratic Programming (MIQCQP) problem. We show that the problem is NP-hard with object placement decision and rendering level selection decision quadratically coupling together. To solve this problem, we propose the Alternating Directions Method of Multipliers (ADMM) algorithm which can iteratively decouple the quadratic terms and reform the problem into the efficiently solvable MIQCQP-1 (i.e., MIQCQP with one constraint) problem. Trace driven simulation shows that our algorithm fits the edge assisted mobile VR game scenario well with fast computation time (at least 4 orders of magnitude less computation time compared to Gurobi solver) and good performance (at least 18% of user visual QoE improvement compared to other mobile VR scheme).

On the Flow Problem in Water Distribution Networks: Uniqueness and Solvers

Increasing concerns on the security and quality of water distribution systems (WDS), call for computational tools with performance guarantees. To this end, this work revisits the physical laws governing water flow (WF) and provides a hierarchy of solvers of complementary value. Given the water injection or pressure at each WDS node, finding the WFs within pipes and pumps along with the pressures at all WDS nodes, constitutes the WF problem. The latter entails solving a set of (non)-linear equations. We extend uniqueness claims on the solution to the WF equations in setups with multiple fixed-pressure nodes and detailed pump models. For networks without pumps, the WF solution is already known to be the minimizer of a convex function. The latter approach is extended to networks with pumps but not in cycles, through a stitching algorithm. For networks with nonoverlapping cycles, a provably exact convex relaxation of the pressure drop equations yields a mixed-integer quadratically constrained quadratic program (MI-QCQP) solver. A hybrid scheme combining the MI-QCQP with the stitching algorithm can handle a WDS with overlapping cycles, but without pumps on them. Each solver is guaranteed to converge regardless of initialization, as numerically validated on a benchmark WDS.

Control of Pervasive Domestic-Scale Inverters for Minimizing Total Feeder Power

This paper proposes a method for studying the potential benefits of reactive power control by domestic-scale inverters, considering network losses, inverter losses, and load-voltage sensitivity. The model is developed as a mixed-integer quadratically constrained quadratic program (MI-QCQP), using a linearization of the unbalanced distribution load flow equations. Networks both with and without on-load tap changers are studied, with the test cases covering both European and North American-style circuits. The use of domestic inverter control is shown to increase benefits compared to conventional tap control by 20%, reducing the total feeder power draw by as much as 0.5% of the feeder load. In contrast, the minimization of either load or losses in isolation is shown to increase the amount of power that a feeder draws in five of the seven circuits. To reduce the communications overhead of the approach, a control scheme is proposed that specifies the reactive power of all inverters on each phase identically; this approach is shown to realise up to 98% of the potential benefits of inverter reactive power control.

Global optimization of large-scale MIQCQPs via cluster decomposition: Application to short-term planning of an integrated refinery-petrochemical complex

Integrated refinery-petrochemical facilities are complex systems that require advanced decision-support tools for optimal short-term planning of their operations. The problem can be formulated as a mixed-integer quadratically constrained quadratic program (MIQCQP), in which discrete decisions select operating modes for the process units, while the entire process network is represented by input-output relationships based on bilinear expressions describing yields and stream properties, pooling equations, fuels blending indices and cost indicators. We develop a novel decomposition-based algorithm for deterministic global optimization that divides the network into small clusters according to their functionality. Inside each cluster, we derive a mixed-integer linear programming (MILP) relaxation based on piecewise McCormick envelopes, dynamically partitioning the variables that belong to the cluster and reducing their domains through optimality-based bound tightening. Results for an industrial case study in Colombia show profit improvements above 10% and significantly reduced optimality gaps compared with the state-of-the-art global optimization solvers BARON and ANTIGONE.

Natural Gas Flow Solvers Using Convex Relaxation

The vast infrastructure development, gas flow (GF) dynamics, and complex interdependence of gas with electric power networks call for advanced computational tools. Solving the equations relating gas injections to pressures and pipeline flows lies at the heart of natural gas network (NGN) operation, yet existing solvers that require careful initialization and uniqueness has been an open question. In this context, this article considers the nonlinear steady-state version of the GF problem. It first establishes that the solution to the GF problem is unique under arbitrary NGN topologies, compressor types, and sets of specifications. For GF setups where pressure is specified on a single (reference) node and compressors do not appear in cycles, the GF task is posed as n convex minimization. To handle more general setups, a GF solver relying on a mixed-integer quadratically constrained quadratic program (MI-QCQP) is also devised. This solver can be used for any GF setup at any NGN. It introduces binary variables to capture flow directions, relaxes the pressure drop equations to quadratic inequality constraints, and uses a carefully selected objective to promote the exactness of this relaxation. The relaxation is probably exact in NGNs with nonoverlapping cycles and a single fixed-pressure node. The solver handles efficiently the involved bilinear terms through McCormick linearization. Numerical tests validate our claims, demonstrate that the MI-QCQP solver scales well, and that the relaxation is exact even when the sufficient conditions are violated, such as in NGNs with overlapping cycles and multiple fixed-pressure nodes.

Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming

We propose methods for improving the relaxations obtained by the normalized multiparametric disaggregation technique (NMDT). These relaxations constitute a key component for some methods for solving nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problems. It is shown that these relaxations can be more efficiently formulated by significantly reducing the number of auxiliary variables (in particular, binary variables) and constraints. Moreover, a novel algorithm for solving MIQCQP problems is proposed. It can be applied using either its original NMDT or the proposed reformulation. Computational experiments are performed using both benchmark instances from the literature and randomly generated instances. The numerical results suggest that the proposed techniques can improve the quality of the relaxations.

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.

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.

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.

Convex relaxations and MIQCQP reformulations for a class of cardinality-constrained portfolio selection problems

In this paper we investigate a class of cardinality-constrained portfolio selection problems. We construct convex relaxations for this class of optimization problems via a new Lagrangian decomposition scheme. We show that the dual problem can be reduced to a second-order cone program problem which is tighter than the continuous relaxation of the standard mixed integer quadratically constrained quadratic program (MIQCQP) reformulation. We then propose a new MIQCQP reformulation which is more efficient than the standard MIQCQP reformulation in terms of the tightness of the continuous relaxations. Computational results are reported to demonstrate the tightness of the SOCP relaxation and the effectiveness of the new MIQCQP reformulation.