Convex Quadratic Relaxation Research Articles

An efficient branch-and-bound algorithm using an adaptive branching rule with quadratic convex relaxation for globally solving general linear multiplicative programs

This article proposes an efficient global algorithm for solving general linear multiplicative programming problem (GLMP). The new algorithm combines the quadratic convex relaxation problem, adaptive branching rule, region reduction technique and branch-and-bound scheme. Firstly, a transformation technique can transform GLMP into a non-convex quadratic program with quadratic and linear constraints. The non-convexity parts of the equivalent problem are addressed by employing secant lines, so that a quadratic convex relaxation problem is structured. Secondly, we introduce the adaptive branching rule to improve the upper bound of the optimal value. Thirdly, the convergence of the proposed algorithm is analyzed and its worst-case complexity is provided. Finally, numerical experiments demonstrate the efficiency and advantage of the proposed algorithm for obtaining the global ɛ-optimal solutions of test instances.

A tight compact quadratically constrained convex relaxation of the Optimal Power Flow problem

In this paper, we consider the Optimal Power Flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consists in a spatial branch-and-bound algorithm based on a compact quadratically-constrained convex relaxation. It is computed by solving the semidefinite rank relaxation of OPF once at the root node of the algorithm. An important result is that the optimal value of our compact relaxation is equal to the rank relaxation value. Then, at every sub-nodes of our branch-and-bound, the lower bound is obtained by solving a quadratic convex problem instead of an SDP. Another contribution is that we add only O(n+m) variables that model the squares of the initial variables, where n is the number of buses in the power system and m the number of transmission lines, to construct our relaxation. Then, since the relations between the initial and auxiliary variables are non-convex, we relax them to get a quadratic convex relaxation. Finally, in our branch-and-bound algorithm, we only have to force a reduced number of equalities to prove global optimality. This quadratic convex relaxation approach is here tailored to the OPF problem, but it can address any application whose formulation is a quadratic optimization problem subject to quadratic equalities and ring constraints. Our first experiments on instances of the OPF problem show that our new algorithm Compact OPF (COPF) is more efficient than the standard solvers and other quadratic convex relaxation based methods we compare it with.

Fast QC Relaxation of the Optimal Power Flow Using the Line-Wise Model for Representing Meshed Transmission Networks

In this paper, we investigate the recently introduced McCormick-based Quadratic Convex (QC) relaxation of the Optimal Power Flow (OPF) where Line-Wise Model (LWM) is used for representing meshed transmission systems (QC-LW OPF) for the sake of decisively determining its relationship to other available convex relaxations in the literature. We also extend the recently introduced convex envelope of the tangent function so it would be suitable for test cases with any voltage angle difference range. A computational study where the recently proposed QC-LW OPF formulation is compared to an equivalent McCormick based QC relaxation of the OPF where Bus Injection Model (BIM) is used for representing meshed transmission systems (QC-BI OPF) is presented in this paper. This computational study was conducted using test cases that belong to different operational categories using 123 test cases from the PGLib-OPF library with a bus size range between 3 up to 6515 buses for the sake of understanding the effect of the change of operating conditions on the quality of solutions obtained using the QC-LW OPF and QC-BI OPF formulations. Results are compared using several metrics that testify to the obtained solution’s quality and the problem’s computational complexity. Comparison of results shows that the QC-LW relaxation neither dominates nor is dominated by the QC-BI relaxation in terms of solution quality. Therefore, it dominates the Second Order Cone (SOC) relaxation and neither dominates nor is dominated by the Semidefinite Programming (SDP) relaxation. Furthermore, it is shown that the QC-LW OPF has reduced the number of relaxed trigonometric functions and McCormick envelopes needed when compared to the QC-BI OPF, leading to a faster solution time for more than 84% of the test cases in the range of 2% up to 67%.

KKT-based primal-dual exactness conditions for the Shor relaxation

In this work we present some exactness conditions for the Shor relaxation of diagonal (or, more generally, diagonalizable) QCQPs, which extend the conditions introduced in different recent papers about the same topic. It is shown that the Shor relaxation is equivalent to two convex quadratic relaxations. Then, sufficient conditions for the exactness of the relaxations are derived from their KKT systems. It will be shown that, in some cases, by this derivation previous conditions in the literature, which can be viewed as dual conditions, since they only involve the Lagrange multipliers appearing in the KKT systems, can be extended to primal-dual conditions, which also involve the primal variables appearing in the KKT systems.

SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs

We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the equality constraints. We show that our relaxations are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a well-known semidefinite program relaxation. The new relaxations are implemented in the global optimization solver BARON, and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that, for our test problems, these relaxations lead to a significant improvement in the performance of BARON.

Co-optimization planning of integrated electricity and district heating systems based on improved quadratic convex relaxation

The rapid growth of combined heat and power (CHP) units has led to the development of integrated electricity and district heating systems (IEHS). To support the design of a highly efficient energy supply system, this paper proposes a long-term co-optimization planning model for an IEHS. Not only CHP units, non-CHP thermal generators, wind farms and electric boilers but also transmission lines and heat pipelines are considered as investment candidates to meet electricity and heat demands. Nonlinear hydraulic conditions and thermal conditions are adopted to precisely capture the characteristics of the heating system. To make the planning model tractable, the nonlinear hydraulic conditions are approximated through piecewise linearization. Based on the introduction of auxiliary variables, the nonconvex thermal conditions are reformulated into linear constraints through quadratic convex relaxation. Hence, the planning model is converted into a large-scale mixed integer linear programming (MILP) problem. Since the planning model is formulated based on independent load blocks, a parallel Benders decomposition algorithm combined with the sequential bound-tightening procedure is proposed to efficiently obtain high-quality solutions. Numerical cases are studied based on two IEHSs of different scales to validate the effectiveness of the proposed co-optimization planning model and the feasibility of the proposed solution methods for solving this complicated planning model for an IEHS.

Quadratic problems with two quadratic constraints: convex quadratic relaxation and strong lagrangian duality

In this paper, we study a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. We introduce two convex quadratic relaxations (CQRs) and discuss cases, where the problem is equivalent to exactly one of the CQRs. Particularly, we show that the global optimal solution can be recovered from an optimal solution of the CQRs. Through this equivalence, we introduce new conditions under which the problem enjoys strong Lagrangian duality, generalizing the recent condition in the literature. Finally, under the new conditions, we present necessary and sufficient conditions for global optimality of the problem.

Spectral Relaxations and Branching Strategies for Global Optimization of Mixed-Integer Quadratic Programs

We consider the global optimization of nonconvex (mixed-integer) quadratic programs. We present a family of convex quadratic relaxations derived by convexifying nonconvex quadratic functions through perturbations of the quadratic matrix. We investigate the theoretical properties of these relaxations and show that they are equivalent to some particular semidefinite programs. We also introduce novel branching variable selection strategies motivated by the quadratic relaxations investigated in this paper. The proposed relaxation and branching techniques are implemented in the global optimization solver BARON and tested by conducting numerical experiments on a large collection of problems. Results demonstrate that the proposed implementation leads to very significant reductions in BARON's computational times to solve the test problems.

Coordination of Electricity, Heat, and Natural Gas Systems Accounting for Network Flexibility

Existing energy networks can foster the integration of uncertain and variable renewable energy sources by providing additional operational flexibility. In this direction, we propose a combined power, heat, and natural gas dispatch model to reveal the maximum potential “network flexibility”, corresponding to the ability of natural gas and district heating pipelines to store energy. To account for both energy transport and linepack in the pipelines in a computational efficient manner, we explore convex quadratic relaxations of the nonconvex flow dynamics of gas and heat. The resulting model is a mixed-integer second-order cone program. An ex-post analysis ensures feasibility of the heat dispatch, while keeping the relaxation of the gas flow model sufficiently tight. The revealed flexibility is quantified in terms of system cost compared to a dispatch model neglecting the ability of natural gas and district heating networks to store energy.

The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.

Convex Relaxations for Quadratic On/Off Constraints and Applications to Optimal Transmission Switching

This paper studies mixed-integer nonlinear programs featuring disjunctive constraints and trigonometric functions and presents a strengthened version of the convex quadratic relaxation of the optimal transmission switching problem. We first characterize the convex hull of univariate quadratic on/off constraints in the space of original variables using perspective functions. We then introduce new tight quadratic relaxations for trigonometric functions featuring variables with asymmetrical bounds. These results are used to further tighten recent convex relaxations introduced for the optimal transmission switching problem in power systems. Using the proposed improvements, along with bound propagation, on 23 medium-sized test cases in the PGLib benchmark library with a relaxation gap of more than 1%, we reduce the gap to less than 1% on five instances. The tightened model has promising computational results when compared with state-of-the-art formulations.

Complexity Results and Effective Algorithms for Worst-Case Linear Optimization Under Uncertainties

In this paper, we consider the so-called worst-case linear optimization (WCLO) with uncertainties on the right-hand side of the constraints. Such a problem often arises in applications such as in systemic risk estimation in finance and stochastic optimization. We first show that the WCLO problem with the uncertainty set corresponding to the [Formula: see text]p-norm ((WCLOp)) is NP-hard for p ɛ (1,∞). Second, we combine several simple optimization techniques, such as the successive convex optimization method, quadratic convex relaxation, initialization, and branch-and-bound (B&B), to develop an algorithm for (WCLO2) that can find a globally optimal solution to (WCLO2) within a prespecified ε-tolerance. We establish the global convergence of the algorithm and estimate its complexity. We also develop a finite B&B algorithm for (WCLO∞) to identify a global optimal solution to the underlying problem, and establish the finite convergence of the algorithm. Numerical experiments are reported to illustrate the effectiveness of our proposed algorithms in finding globally optimal solutions to medium and large-scale WCLO instances.

Robust linear model for multi‐period AC optimal power flow

In the near future power systems, efficient management of uncertainties with considering the system constraints without any simplification will be a challenge for system operators. Considering AC constraints leads to providing more accurate schedule of generating units, which can have a significant impact on the reduction of operating costs. Although numerous studies have been done to convexify AC optimal power flow constraints, most of the models are non-linear, which can be intractable for large-scale systems. In this study, a novel linear robust AC model is introduced using a combination of the quadratic convex relaxation (QCR) and the Frank–Wolfe algorithm for linearising the AC constraints. The uncertainties are modelled by applying the robust optimisation with recourse to obtain an optimal schedule for the conventional units in multi-period real-time markets. The Benders-dual algorithm is implemented to solve the optimisation problem. The proposed model was applied to the IEEE 3-bus, 118-bus, and 300-bus systems. The results indicate that the proposed algorithm obtains more precise approximation than the QCR method. In addition, the costs and losses of the proposed model are less than those of the conventional robust DC and stochastic models. Furthermore, because the proposed model is linear, its runtime is rational.

Globally solving nonconvex quadratic programming problems with box constraints via integer programming methods

We present effective linear programming based computational techniques for solving nonconvex quadratic programs with box constraints (BoxQP). We first observe that known cutting planes obtained from the Boolean Quadric Polytope (BQP) are computationally effective at reducing the optimality gap of BoxQP. We next show that the Chvatal–Gomory closure of the BQP is given by the odd-cycle inequalities even when the underlying graph is not complete. By using these cutting planes in a spatial branch-and-cut framework, together with a common integrality-based preprocessing technique and a particular convex quadratic relaxation, we develop a solver that can effectively solve a well-known family of test instances. Our linear programming based solver is competitive with SDP-based state of the art solvers on small instances and sparse instances. Most of our computational techniques have been implemented in the recent version of CPLEX and have led to significant performance improvements on nonconvex quadratic programs with linear constraints.

Strengthening the SDP Relaxation of AC Power Flows With Convex Envelopes, Bound Tightening, and Valid Inequalities

This work revisits the semidefine programming (SDP) relaxation of the ac power flow equations in light of recent results illustrating the benefits of bounds propagation, valid inequalities, and the convex quadratic relaxation. By integrating all of these results into the SDP model a new hybrid relaxation is proposed, which combines the benefits from all of these recent works. This strengthened SDP formulation is evaluated on 71 AC Optimal Power Flow test cases from the NESTA archive and is shown to have an optimality gap of less than 1% on 63 cases. This new hybrid relaxation closes 50% of the open cases considered, leaving only eight for future investigation.

Convex quadratic relaxations for mixed-integer nonlinear programs in power systems

This paper presents a set of new convex quadratic relaxations for nonlinear and mixed-integer nonlinear programs arising in power systems. The considered models are motivated by hybrid discrete/continuous applications where existing approximations do not provide optimality guarantees. The new relaxations offer computational efficiency along with minimal optimality gaps, providing an interesting alternative to state-of-the-art semidefinite programming relaxations. Three case studies in optimal power flow, optimal transmission switching and capacitor placement demonstrate the benefits of the new relaxations.

The QC Relaxation: A Theoretical and Computational Study on Optimal Power Flow

Convex relaxations of the power flow equations and, in particular, the semi-definite programming (SDP) and second-order cone (SOC) relaxations, have attracted significant interest in recent years. The quadratic convex (QC) relaxation is a departure from these relaxations in the sense that it imposes constraints to preserve stronger links between the voltage variables through convex envelopes of the polar representation. This paper is a systematic study of the QC relaxation for AC optimal power flow with realistic side constraints. The main theoretical result shows that the QC relaxation is stronger than the SOC relaxation and neither dominates nor is dominated by the SDP relaxation. In addition, comprehensive computational results show that the QC relaxation may produce significant improvements in accuracy over the SOC relaxation at a reasonable computational cost, especially for networks with tight bounds on phase angle differences. The QC and SOC relaxations are also shown to be significantly faster and reliable compared to the SDP relaxation given the current state of the respective solvers.

Relaxing Nonconvex Quadratic Functions by Multiple Adaptive Diagonal Perturbations

The current bottleneck of globally solving mixed-integer (non-convex) quadratically constrained problem (MIQCP) is still to construct strong but computationally cheap convex relaxations, especially when dense quadratic functions are present. We propose a cutting surface procedure based on multiple diagonal perturbations to derive strong convex quadratic relaxations for nonconvex quadratic problem with separable constraints. Our resulting relaxation does not use significantly more variables than the original problem, in contrast to many other relaxations based on lifting. The corresponding separation problem is a highly structured semidefinite program (SDP) with convex but non-smooth objective. We propose to solve this separation problem with a specialized primal-barrier coordinate minimization algorithm. Computational results show that our approach is very promising. First, our separation algorithm is at least an order of magnitude faster than interior point methods for SDPs on problems up to a few hundred variables. Secondly, on nonconvex quadratic integer problems, our cutting surface procedure provides lower bounds of almost the same strength with the diagonal SDP bounds used by (Buchheim and Wiegele, 2013) in their branch-and-bound code Q-MIST, while our procedure is at least an order of magnitude faster on problems with dimension greater than 70. Finally, combined with (linear) projected RLT cutting planes proposed by (Saxena, Bonami and Lee, 2011), our procedure provides slightly weaker bounds than their projected SDP+RLT cutting surface procedure, but in several order of magnitude shorter time. Finally we discuss various avenues to extend our work to design more efficient branch-and-bound algorithms for MIQCPs.

Strong SOCP Relaxations for the Optimal Power Flow Problem

This paper proposes three strong second order cone programming (SOCP) relaxations for the AC optimal power flow (OPF) problem. These three relaxations are incomparable to each other and two of them are incomparable to the standard SDP relaxation of OPF. Extensive computational experiments show that these relaxations have numerous advantages over existing convex relaxations in the literature: (i) their solution quality is extremely close to that of the standard SDP relaxation (the best one is within 99.96% of the SDP relaxation on average for all the IEEE test cases) and consistently outperforms previously proposed convex quadratic relaxations of the OPF problem, (ii) the solutions from the strong SOCP relaxations can be directly used as a warm start in a local solver such as IPOPT to obtain a high quality feasible OPF solution, and (iii) in terms of computation times, the strong SOCP relaxations can be solved an order of magnitude faster than the standard SDP relaxation. For example, one of the proposed SOCP relaxations together with IPOPT produces a feasible solution for the largest instance in the IEEE test cases (the 3375-bus system) and also certifies that this solution is within 0.13% of global optimality, all this computed within 157.20 seconds on a modest personal computer. Overall, the proposed strong SOCP relaxations provide a practical approach to obtain feasible OPF solutions with extremely good quality within a time framework that is compatible with the real-time operation in the current industry practice.

Exact quadratic convex reformulations of mixed-integer quadratically constrained problems

We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.