1

ABSTRACT—

AC Optimal Power Flow problem is a large-scale and nonlinear programming with a nonconvex feasible region. Seeking for

the global optimizer has attracted much attention recently. This paper presents an efficient technique to find the global optimal

solution by the Divide-and-Conquer type approach. Convex relaxation of AC OPF in the form of semi-definite programming was

performed to find a lower bound for the objective function value at the global solution. The feasible region of OPF was divided

into multiple sub-regions with cuts based upon voltage magnitude and voltage angle constraints. The constraints proposed were

developed after the projection of the decision variable of the relaxed problem onto the feasible region of the original problem when

the relaxed problem produced a solution that violated the rank-1 assumption. Several IEEE model systems were used in this paper

to illustrate the effectiveness of the proposed method. Multiple feasible solutions were identified and the global optimal solution

with a small gap was found. Challenges in computation costs are discussed and the parallelization of the algorithm to reduce the

computation costs is also presented.

Keywords-

Power System Analysis Computing, Power System Economics, Power System Simulation, Mathematical Programming

1 Gokturk Poyrazoglu is currently with Electric Power Research Institute (EPRI), Charlotte, NC USA.

Efficient AC Optimal Power Flow

and Global Optimizer Solutions

Gokturk Poyrazoglu1, HyungSeon Oh2,3

1 The Department of Electrical Engineering, University at Buffalo, Buffalo, NY 14260 USA,

gokturkp@buffalo.edu

2 The Department of Electrical Engineering, University at Buffalo, Buffalo, NY 14260 USA,

hso1@buffalo.edu

3 Corresponding author

2

I. NOMENCLATURE

C Summation of generation cost

N Number of buses

L Number of transmission line

i, l Bus index and line index

xi Real part of complex voltage at Bus i

yi Imaginary part of complex voltage at Bus i

p , p Min. and max. limits of real power generation

q , q Min. and max. limits of reactive power generation

dP Vector of real power demand

dQ Vector of reactive power demand

gp Function of real power injection

gq Function of reactive power injection

f , lf Vector of maximum power flow on transmission line and maximum power flow on Line l

flow Function of power flow on transmission line

V Function of voltage magnitude

E , E Vector of voltage magnitude limits

iE , iE Min. and max. voltage magnitude limits at Bus i

W, W* Matrix variable in SDP and the optimal solution

θi, iθ , iθ Voltage angle and min. and max. limits at Bus i

Eb, θb Dividing point of voltage magnitude and angle

ΔOPF Violation of OPF constraints

heq, hineq General form of equality and inequality constraints

λ Vector of eigenvalue of matrix W

λ1, λ2 Largest and second largest eigenvalue of matrix W

vEVD Approximated vector of real and imaginary voltage for eigenvalue decomposition of matrix W*

ΔOPF Violation of rank-1 condition

SFTB So-Far-The-Best (SFTB) solution

3

II. INTRODUCTION

Since Carpentier [1] introduced the optimal power flow (OPF) problem in 1962, several techniques have been developed to

solve this non-linear and nonconvex problem [2-5]. OPF finds an optimal operating cost solution for power generation and voltage

variables while satisfying Kirchhoff’s laws, operation constraints, and economic, policy, and/or environmental constraints. Due to

its key role in the operation and planning of power systems operations, both industry and academia have given great attention to

the OPF problem. The nonlinearity and the large scale of the problem make it practically impossible to find the global solution in

real-time, and therefore, heuristic approaches have focused on finding a local numerical solution [6-10].

Various attempts have been made to determine the global optimizer, including the use of stochastic and probabilistic methods

[11-13], such as simulated annealing [14], direct Monte-Carlo sampling [15], mean field theory [16], and deterministic methods,

including cutting plane [17, 18], Branch-and-Bound (B&B) [19-20], and Branch-and-Cut [21]. However, the non-convex feature

has still remained problematic. To address this issue, Bai [22, 23] introduced the conversion of the OPF problem into semi-definite

programming (SDP). Thereafter, Lavaei and Low [24] showed that the solution of SDP-OPF could be the global optimizer of the

original OPF problem in the form of non-linear programming, if and only if, the solution satisfies a rank requirement. Molzahn

and Lesieutre [25] later presented a case showing that the solution in [24] did not satisfy the rank requirement. The dissatisfaction

of rank requirement is also problematic in Second Order Cone Programming (SOCP) as a relaxation of OPF [46]. More case studies

to support the foundation of [25] may be found in [26]. Since then, several studies have demonstrated methods to find satisfactory

conditions that provide the global optimizer of OPF [27-33]. Nonetheless, the only feature that has been proven invariably correct

for SDP-OPF is that its convex feasible region always covers the nonconvex feasible region of the original OPF, and its objective

function value is always less than or equal to the global objective function value of the original OPF [22,24].

Several papers have proposed methods to address cases with high-rank solutions. Josz [34] modified rank relaxation and found

the rank-satisfied solutions. However, the method cannot handle cases larger than the 5-bus system because of high runtime costs.

Gopalakrishnan [20] bisected the region of voltage magnitude, and real and reactive power, and resolved SDP in the sub-region.

Thus, feasible solutions are derived from a nonlinear solver.

We pursued the global optimizer by dividing the feasible region of the original OPF with the “Divide-and-Conquer” (D&C)

method. This paper’s first contribution is the introduction of a novel angular cut for OPF with which to divide the nonconvex

feasible region. When the entire feasible region is covered by the D&C method, the best solution is guaranteed to be the global

optimizer of the original OPF problem. Thus, rather than using a nonlinear solver, the method proposed was able to determine

several rank-satisfied solutions from the feasible region of OPF by solving SDP problems within the sub-regions. Rather than

solving a Lagrangian dual problem [19, 20], solving SDP not only provided a lower bound, but also find the feasible solutions for

the OPF problem. The most striking feature of our approach is the capability to guarantee the infeasibility if the original problem

4

does not have any feasible solution. Our second contribution is the demonstration of a method to divide the feasible region by the

D&C method. Instead of bisecting feasible region, in this paper, the branching point was derived from the SDP solution of the

parent node to guarantee that the solution of the parent node did not repeat in the child nodes. Thereafter, we modified the bounds

in the child nodes to achieve a narrow feasible region.

III. PROBLEM FORMULATION

The OPF problem considered in this study can be modeled as (P1) [35] where p and q are the vectors collect real and reactive

power generation respectively, x and y are the vectors collect real and imaginary part of complex voltage respectively:

(P1) ( ), , ,minx y p q

C p (1a)

subject to

( ), , Ppg x y p d= (1b)

( ), , Qqg x y q d= (1c)

( ) 2,flow x y f≤ (1d)

( ),E V x y E≤ ≤ (1e)

p p p≤ ≤ (1f)

q q q≤ ≤ (1g)

The objective is to minimize a convex quadratic cost function that depends on real power generation. The constraint set includes

real and reactive power balance equations (1b) and (1c), respectively. The left hand side of (1b) and (1c) are in general nonlinear

quadratic functions of optimization variables (x, y, p, and q). The power balance equations lead to a non-convex feasible region.

Later, the transformation of (P1) to SDP may be used to relax the non-convexity of OPF [24]. The constraint set of (P1) also

includes the maximum bound on power flow (flow), the minimum and maximum bounds on voltage magnitude ( E and E ), real

and reactive power generations ( p and p , q and q ).

(P2) ( ), ,minW p q

C p (2a)

subject to:

( ) 1...P ii i pp d tr W i N− = Φ ∀ = (2b)

( ) 1...Q ii i qq d tr W i N− = Φ ∀ = (2c)

5

( )22 1...iiiE tr W E i N≤ Π ≤ ∀ = (2d)

p p p≤ ≤ (2e)

q q q≤ ≤ (2f)

− fl2

tr Ψ plW( ) tr Ψq

lW( )tr Ψ p

lW( ) −1 0

tr ΨqlW( ) 0 −1

#

$

%%%%%%

&

'

((((((

0 ∀l =1…L (2g)

W0; (2h)

(P2) is in the form of SDP, where N is the number of buses, L is the number of transmission lines, and matrices Φ, Π, and Ψ are

defined in Appendix A. The feasible region of (P2) is convex and it is a relaxation of (P1). If the solution of (P2) satisfies the rank

requirement for the matrix variable W (rank (W) = 1), then a unique vector v, which is constructed from the vectors x and y (v=[x;y]),

may be carried out by W=vvT. Thus, the related vector v becomes the global optimizer of the original problem (P1) [24]. However,

if (P2) gives a high-rank solution for W, the solution of (P2) yields only a lower bound for (P1), and the optimizer does not have a

physical meaning for the power systems [25].

In order to find a meaningful solution out of high rank result, D&C method is adopted in this study. When the feasible region

is divided into sub-regions and when one of the sub-regions becomes sufficiently small, non-convexity does not affect optimality

significantly, i.e., there can exist a rank-1 solution in the sub-region. The method proposed in this study as a way to narrow the

feasible region is to introduce a new box constraint on the voltage angle and to modify the bounds on the voltage magnitude.

Specifically, the proposed algorithm updates the minimum and maximum limits for voltage magnitude and inserts a box constraint

on W for the voltage angle. The derivation of the additional constraint is as follows:

The voltage angle at Bus i may be defined as below.

1tan ii

i

yx

θ − ⎛ ⎞= ⎜ ⎟

⎝ ⎠ (3)

We defined the maximum and minimum voltage angle constraints as iiiθ θ θ≤ ≤ .The tangent function of an angle, tan(θi), is

monotonic over the interval (-90 ,90). This is indeed true for power systems operations, because we operated the system with a

low angle to avoid instability in the system. In the situation in which θi may exceed 90 degrees in absolute value, we redefined the

angle region into (-270, -90), (-90, 90) and (90, 270) degrees. In each, tan(θi) remained monotonically increasing or decreasing.

The maximum and minimum voltage angle constraints can be written as Eqs. (4a) to (4b) when xi is positive:

6

yi − xi tan θi( ) ≥ 0 (4a)

yi − xi tan θi( ) ≤ 0 (4b)

or Eqs. (4c) and (4d) when xi is negative:

yi − xi tan θi( ) ≤ 0 (4c)

yi − xi tan θi( ) ≥ 0 (4d)

An equivalent constraint on the voltage angle at Bus i can be defined as in Eq. (5), and converted into the SDP problem, as in

Eq. (6):

yi − xi tan θi( )( ) yi − xi tan θi( )( ) ≤ 0 (5)

Wi+N ,i+N − tanθi + tanθi( )Wi ,i+N +Wi ,i tanθi tanθi ≤ 0 (6)

By modifying the voltage angle and magnitude bounds, a new problem may be formulated, as shown in (P3). This problem is

solved at all child nodes.

(P3) minW ,p,q

C p( ) (7a)

subject to

(2b) through (2h)

Wi+N ,i+N − tanθi + tanθi( )Wi ,i+N +Wi ,i tanθi tanθi ≤ 0

∀i =1N (7b)

Consider W* as the optimal solution of (P2) or (P3). If the rank of W* is greater than one, the approximate solution vector, vEVD,

can be obtained by eigenvalue decomposition of W*, as given in Eq. (8b) where λ1 is the largest eigenvalue of W*.

* TW UDU= (8a)

1 1EVDv uλ= (8b)

The voltage magnitude and angle at Bus i are in the space of W*, and therefore accordingly of vEVD. However, the voltage

magnitude and angle from vEVD may not be feasible. Therefore, vEVD cannot be used to create new boundaries of (P3) for a new

child node. As shown in Eqs. (9a) and (9b), rather than using vEVD, we created and then branched on voltage magnitudes and angles

from W*:

7

Eib =Ei + Ei2

(9a)

θib =12tan−1

Wi ,i+N* tanθi −Wi+N ,i+N

*

Wi ,i* tanθi −Wi ,i+N

*+ tan−1

Wi ,i+N* tanθi −Wi+N ,i+N

*

Wi ,i* tanθi −Wi ,i+N

*

"

#

$$

%

&

''

(9b)

By choosing a voltage angle bound, θib, and a voltage magnitude bound, Ei

b, new child nodes are created by updating Eqs. (2d)

and (7b), as shown in Eqs. (10a) to (10h):

Node 1 Ei2≤ tr ΠiW( ) ≤ Ei

b( )2

(10a)

Wi+N ,i+N − tanθi + tanθib( )Wi ,i+N +Wi ,i tanθi tanθi

b ≤ 0 (10b)

Node 2 Eib( )2≤ tr ΠiW( ) ≤ Ei

2 (10c)

Wi+N ,i+N − tanθi + tanθib( )Wi ,i+N +Wi ,i tanθi tanθi

b ≤ 0 (10d)

Node 3 Ei2≤ tr ΠiW( ) ≤ Ei

b( )2

(10e)

Wi+N ,i+N − tanθib + tanθi( )Wi ,i+N +Wi ,i tanθi

b tanθi ≤ 0 (10f)

Node 4 Eib( )2≤ tr ΠiW( ) ≤ Ei

2 (10g)

Wi+N ,i+N − tanθib + tanθi( )Wi ,i+N +Wi ,i tanθi

b tanθi ≤ 0 (10h)

By following Eq.(10a) through Eq.(10h), we guarantee that θib and Ei

b lie in the box, and the solution of the parent node (W*)

is infeasible in all child nodes (see Appendix B).

If the constraints of (P1) are written in general form as heq = 0, hineq ≤ 0, then the violation of constraints may be defined as

below:

ΔOPF = heq max hineq ,0( )"#$

%&'

T

(11)

If ||ΔOPF|| ≤ 10-5, we claim that vEVD is a feasible solution for (P1). The definition of the violation of rank-1 condition, ΔEVD, is

as follows:

ΔEVD = λ2 λ1 (12)

Flowchart for the proposed method is shown in Fig.1. The SFTB solution was initialized as positive infinite, and the minimum

and maximum angles of the root node were initialized as -90 and 90 degrees. The SDP problem given in (P3) was solved once at

the root node in the branch tree. Then, the feasible region of OPF was divided into parts with voltage magnitude and angular

8

constraints. Four SDP problems in the form of (P3) were created for four child nodes with feasible sub-regions. If the solution of

(P3) at a child node satisfied the condition given in Eq. (13), then further exploration of that region was not worthwhile. Therefore,

we marked that region as solved and did not further branch that region:

( )( 3) 1 objObjFncVal P SFTBε≥ − (13)

where εobj is the error bound of the objective function. Efficiency is further improved by converting Eq. (13) into an additional

constraint and by imposing on P3 as follows:

C p( ) ≤ 1−εobj( ) SFTB (14)

If this constraint is violated, then the node is not worthwhile to investigate, and therefore, the node is terminated.

Fig.1. Flowchart of Divide-and-Conquer method

9

If the objective function value of (P3) was lower than SFTB, then we solved the voltage vector v. If the resultant vector v was

feasible for the original OPF, then we updated SFTB and denoted the node as solved and feasible; otherwise, the region deserved

further investigation, in which case we divided the region again by modifying the constraints obtained from the solution of Eib and

θib. In each step, the node with smallest ΔEVD is chosen to be solved, in order to make the matrix closer to rank-1. If there was no

unsolved node left in the branch tree, and an SFTB was updated, then we terminated the process and claimed that SFTB was the

global solution with the εobj gap. At the end of the proposed algorithm, if all nodes were infeasible, then we claimed that the original

problem was guaranteed to be infeasible.

IV. NUMERICAL EXAMPLE

In order to demonstrate the effectiveness of the proposed method, a modified IEEE 14-bus system (hereinafter Case 14A) is

adopted for a numerical example. Case14A was first studied in Ref. [36] and the MATPOWER suitable case file may be found

online at reference [45]. The system has five generators and twenty transmission lines. baseMVA is assumed to be 100MVA

because power systems are modeled in per unit (p.u.) bases [37]. Lastly, the minimum real power generation of all generators set

to be zero.

A nonlinear programming solver, IPOPT [38], is used to solve the problem. A local optimal objective function value of the

problem (P1) is 9093.7$/hr, with a real power loss of 3.72MW. The following constraints associated with one reactive power

generation, one real power generation, seven power flow limits, and one voltage limit are active at the local solution. The SDP

relaxation of (P1), as given in (P2), is solved for the same system in order to show the dissatisfaction of rank requirement at the

local solution found by MATPOWER. A modeling tool, YALMIP [40], is used in MATLAB to model (P3), and the SeDuMi 1.3

[41] is used as a solver. The optimal solution of (P2) resulted in an objective function value of 8,092.8$/hr with rank (W) = 3. As

discussed in Section II, a high rank solution of (P2) is not feasible to the original OPF (P1), and can only be used as a lower bound

of the objective function value. However, the approximate rank-1 matrix to W* can be obtained by eigenvalue decomposition of

W*, as given in Eq.(8a), leading to an approximate solution vector, vEVD, as shown in Eq.(8b), where λ are the eigenvalues of W.

Three nonzero eigenvalues are given in (15). The approximate solution vector, vEVD, for (P2) is presented in Table I and the solution

of the power generation variables are shown in Table II.

[ ]14.5605; 0.0044; 0.0040λ = (15)

The two solutions gathered from an SDP solver and a nonlinear programming solver suggest that the global optimal objective

function value is in the interval between 8,092.8$/hr and 9,093.7$/hr. Once the interval is identified, the proposed D&C algorithm

(Fig. 1) is began. For Case14A, the global solution found to be 8,092.8$/hr. The difference in the values of SDP solver and D&C

10

method is 0.022$/hr. The values of variables of (P2) and the proposed D&C method are all same but the reactive power generation

at the Generator 5.

TABLE I. VOLTAGE SOLUTIONS

Bus Idx (P1) (P2) D&C Method

E θ E θ E θ

1 1.043 0.000 1.060 0.000 1.060 0.000

2 1.036 -0.0236 1.043 -0.0705 1.043 -0.0708

3 1.040 -0.0577 1.017 -0.1731 1.017 -0.1738

4 1.014 -0.0902 1.017 -0.1488 1.006 -0.1491

5 1.013 -0.0771 1.009 -0.1258 1.009 -0.1261

6 1.047 -0.1526 1.027 -0.2111 1.027 -0.2115

7 1.043 -0.1414 1.023 -0.1995 1.023 -0.1999

8 1.060 -0.1389 1.032 -0.1943 1.027 -0.1954

9 1.038 -0.1699 1.019 -0.2294 1.018 -0.2298

10 1.032 -0.1721 1.012 -0.2316 1.012 -0.2320

11 1.036 -0.1648 1.016 -0.2239 1.016 -0.2243

12 1.032 -0.1684 1.012 -0.2275 1.012 -0.2279

13 1.027 -0.1703 1.008 -0.2295 1.007 -0.2299

14 1.015 -0.1882 0.995 -0.2483 0.995 -0.2487

TABLE II. POWER GENERATION SOLUTIONS

Gen Idx (P1) (P2) D&C Method

p q p q p q

1 80.878 0.0000 192.49 0.000 192.49 0.000

2 69.670 4.3239 36.504 39.492 36.504 39.493

3 100.00 29.553 29.546 29.304 29.546 29.303

4 10.599 3.3401 7.1665 -2.2458 7.1664 -2.2458

5 1.5679 10.041 2.6947 7.5837 2.6947 2.8080

11

The proposed D&C method visited 101 nodes to determine the global solution that satisfied the rank requirement of SDP-OPF

with violation of 623.7 10OPF

−Δ = × , and the computation time is 138.27 seconds using Intel® Xeon™ 2.50GHz, 6-core CPU. It

is noteworthy to mention that bi-section of angle and magnitude does not return a solution even after more than 10,000 iterations.

Compared to the bisection method, the method we used achieved the global solution in an efficient manner by dividing regions

with reasonable voltage magnitude and angle constraints, by finding the ideal node at which to branch-out, and by terminating a

node in the branch tree efficiently. Fig. 2 shows the improvement of the SFTB and the lower bound in the D&C method during the

solution process. It can be seen from the figure that two feasible solutions to the original OPF are found by the method proposed,

both of which satisfied the rank requirement of SDP, and are feasible and better solutions than the solution provided by

MATPOWER.

Fig. 2. Algorithm performance for case14A

Table III summarizes the power system networks for testing the D&C Method. Precise system descriptions can be obtained from

(Case3 [25]; Case6ww, Case14, Case30 [39]). The maximum iterations set as 10,000. In each case, the D&C method found multiple

rank-1 solutions. For Case3 and Case6ww, the D&C method achieves a 1% gap after 10,000 iterations, while for Case14 and

Case30, a 0.01% gap was achieved. The runtime for these cases is much longer than for Case14A, because of larger distance

between SDP solutions at the root node and the global solutions.

12

TABLE III. RESULTS OF TEST SYSTEMS

System Lower bound SFTB Iter. Time Gap MATPOWER

Case3 5,790.5 5,823.2 10,000 2,131.2 5.6×10-3 5,812.6

Case6ww 3,132.2 3,144.0 10,000 2,805.3 3.8×10-3 3,144.0

Case14 8,081.3 8,082.1 2,373 1,302.4 9.9×10-5 8,081.5

Case30 576.8 577.1 1,605 104,873 5.4×10-5 576.89

The computational time and the optimality gap are also reported for modified test cases that are generated by randomly

perturbing demand information similar to Refs. [47, 48, 49]. Those results are presented in Table IV, in which the optimality gap

is small that no other global optimal methods achieve such a small gap regardless the system situation (voltage or congestion). The

computational times of those systems are also presented in Fig 3, in which the linear regression line is included only for a visual

guidance.

TABLE IV. RESULTS OF PERTURBED TEST SYSTEMS

System Computation Time (sec) Gap

Case6ww 0.25 1e-5

Case9 15.99 1e-5

Case14 0.86 1e-5

Case30 64.32 1e-5

Case57 13.42 1e-5

Case118 900 0.004

Fig. 3. Computational time of perturbed test systems

13

V. DISCUSSION

The method proposed in this study considers the entire feasible region of the original OPF problem and finds the global optimizer

by dividing the region based on voltage magnitude and angle constraints. Although the theoretical background of the algorithm is

solid and accurate, we observed the computational time of the process is rather high. Although the theoretical background for

different SDP solution techniques is presented in [42-44], the implementation of the solvers is not yet robust and rapid. Second, in

order to cover the entire search space, the method had to visit many nodes. The case study presented in this paper was selected

intentionally to give a high rank solution for (P3), and therefore the constraint set was developed to have a highly nonconvex search

space that required visiting many nodes. To reduce the computation costs, we suggest parallelization of the algorithm, as we

pursued the D&C method.

VI. CONCLUSIONS

The OPF problem plays a key role in operations of power systems. Nonlinear and nonconvex features of OPF make the problem

difficult for a global optimizer to solve. Several methods have been introduced in the literature to find a local optimal solution;

however, recent developments in SDP relaxation of OPF show promise in the determination of the global solution. This paper

focused on identifying physically meaningful data from the high-rank solution matrix of SDP-OPF, and using the information to

divide the feasible region with box constraints on voltage magnitude and voltage angle. By itself, the proposed algorithm solve

SDP problems and efficient methods are presented to fathom the nodes in the branch tree.

IEEE model systems are tested to demonstrate the performance of the algorithm in seeking the global optimizer of OPF problem.

In addition to the global solution, the proposed method may also identify several feasible solutions if exist that have better objective

function value than the solution obtained from the nonlinear programming solver. The proposed flowchart is presented and the

results of several IEEE model systems are shared.

14

APPENDIX A

Let Y represent the admittance matrix of the transmission network. yl denotes the mutual admittance for line l, while lf and lt are

the from-bus and to-bus of line l, respectively. Denote standard basis vector as e1,e2,…,eN. Matrices Φ, Ψ, Π may be defined with

Eqs. (11a) to (11g) for each Bus i and each Line l.

: Ti i iY e e Y= (15a)

Yl := yl + yl( )elf elfT − yl( )elteltT (15b)

{ } { }{ } { }

Re Im1:2 Im Re

T Ti i i ii

p T Ti i i i

Y Y Y Y

Y Y Y Y

⎡ ⎤+ −Φ = ⎢ ⎥

− +⎢ ⎥⎣ ⎦ (15c)

{ } { }{ } { }

Im Re1:2 Re Im

T Ti i i ii

q T Ti i i i

Y Y Y Y

Y Y Y Y

⎡ ⎤+ −Φ = − ⎢ ⎥

− +⎢ ⎥⎣ ⎦ (15d)

{ } { }{ } { }

Re Im1:2 Im Re

T Tl l l ll

p T Tl l l l

Y Y Y Y

Y Y Y Y

⎡ ⎤+ −Ψ = ⎢ ⎥

− +⎢ ⎥⎣ ⎦ (15e)

{ } { }{ } { }

Im Re1:2 Re Im

T Tl l l ll

q T Tl l l l

Y Y Y Y

Y Y Y Y

⎡ ⎤+ −Ψ = − ⎢ ⎥

− +⎢ ⎥⎣ ⎦ (15f)

0:

0

Ti i

i Ti i

e ee e

⎡ ⎤Π = ⎢ ⎥

⎣ ⎦ (15g)

APPENDIX B

Because W* is feasible for P2, it satisfies Eqs. (2h) and (7b),

Wi+N ,i+N* − tanθi + tanθi( )Wi ,i+N

* +Wi ,i* tanθi tanθi ≤ 0 (16a)

W *0; (16b)

From the semi-positive definite property, Eq. (16b), we have,

*, 0i iW ≥ (16c)

* * * 2, , , 0i i i N i N i i NW W W+ + +- ≥ (16d)

Therefore, (16a)×(16c)-(16d) leads to Eq.(16e), which can be converted into Eqs. (16f) and (16g),

Wi ,i+N* −Wi ,i

* tanθi( ) Wi ,i+N* −Wi ,i

* tanθi( ) ≤ 0 (16e)

15

Wi ,i* tanθi −Wi ,i+N

* ≤ 0 (16f)

Wi ,i+N* −Wi ,i

* tanθi ≤ 0 (16g)

From Eq. (9b), we know that,

Wi ,i+N* tanθi −Wi+N ,i+N

*

Wi ,i* tanθi −Wi ,i+N

*< tanθi

b <Wi ,i+N

* tanθi −Wi+N ,i+N*

Wi ,i* tanθi −Wi ,i+N

*

(16h)

Because of Eqs. (16f) and (16g), Eq. (16h) can be converted into Eqs. (16i) and (16j),

Wi+N ,i+N* −Wi ,i+N

* tanθi + tanθib( )+Wi ,i

* tanθi tanθib > 0

(16i)

( )* * *, , ,tan tan tan tan 0b b

i N i N i i N i i i i i iW W Wθ θ θ θ+ + +− + + > (16j)

Therefore, W* violates Eqs. (10b), (10d), (10f) and (10h), and is infeasible for all four child nodes. And (16a)/(16f) and

(16a)/(16g) can get to Eqs. (16k) and (16m),

tanθi ≤Wi ,i+N

* tanθi −Wi+N ,i+N*

Wi ,i* tanθi −Wi ,i+N

*< tanθi

b (16k)

tanθib <Wi ,i+N

* tanθi −Wi+N ,i+N*

Wi ,i* tanθi −Wi ,i+N

*≤ tanθi (16m)

which means that θib lies in the box.

16

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