Constructing QCQP Instances Equivalent to Their SDP Relaxations
Masakazu Kojima
Department of Data Science for Business Innovation (kojima@is.titech.ac.jp).Naohiko Arima
(naoarima@me.com).Sunyoung Kim
Department of Mathematics, Ewha W. University, 52 Ewhayeodae-gil, Sudaemoon-gu, Seoul 03760, Korea
(skim@ewha.ac.kr).
The research was supported by NRF 2021-R1A2C1003810.
Abstract
General quadratically constrained quadratic programs (QCQPs)
are challenging to solve as they are known to be NP-hard.
A popular approach to approximating QCQP solutions is to use
semidefinite programming (SDP) relaxations.
It is well-known that the optimal value of the
SDP relaxation problem bounds the optimal value
of the QCQP from below, i.e., .
The two problems are considered equivalent if .
In the recent paper by Arima, Kim and Kojima [arXiv:2409.07213],
a class of QCQPs that are equivalent to their SDP relaxations
are proposed with no condition imposed on the quadratic objective function, which
can be chosen arbitrarily.
In this work, we explore the construction of
various QCQP instances
within this class to complement the results in [arXiv:2409.07213].
Specifically, we first construct QCQP instances with two variables and then extend them to
higher dimensions. We also discuss how to compute an optimal QCQP solution from the SDP relaxation.
We study a quadratically constrained quadratic program (QCQP) that minimizes a
quadratic objective function in multiple real variables
over a feasible region represented
by quadratic inequalities in the variables.
In general, QCQPs are NP-hard [10]
and become increasingly difficult to solve as the number of variables grows.
It is well-known that
the optimal value of a QCQP is bounded by the optimal value of its SDP
relaxation from below [11, 12].
This property has frequently been utilized
for (approximately) solving the QCQP since the SDP can be solved numerically. We say that the QCQP and
the SDP relaxation are equivalent if .
In [3], conditions on the feasible region
were presented to ensure the equivalence of
a QCQP and its SDP relaxation.
We first illustrate their conditions using the following figure.
The unshaded area, including its boundary,
represents the feasible region where
two variables and can vary without limitation,
while the green ellipsoidal regions indicate the restricted zones labeled
.
The feasible region is enclosed by the ellipsoid 8 within which 7 restricted zones
are positioned. We regard the area outside of the ellipsoid 8 as a restricted zone.
Given a quadratic function in 2 variables , say
, we consider the
optimization problem of minimizing over the feasible region.
The essential properties of this optimization problem are:
(a) The objective function is a quadratic function in multiple variables.
(b) Each restricted zone is represented by a single quadratic inequality
in multiple variables.
(c) Two distinct restricted zones could intersect with their boundaries but their interior never overlap
(a more rigid condition is given as Condition (B)’ below).
Properties (a) and (b) imply that this problem is a
QCQP, while (c) represents a condition imposed on the
QCQPs that we consider throughout the paper.
Since (c) imposes restrictions only on the constraints of QCQPs, the quadratic objective function can be chosen arbitrarily.
It was shown in [3] that every QCQP satisfying (c)
is equivalent to its SDP relaxation.
The study in [3], however, focused mainly on the theoretical relationships conditions,
including (c) that ensure the equivalence of QCQPs and their SDP relaxations, but
did not fully address a variety of QCQP instances satisfying (c), nor
the construction of such instances.
This paper aims to complement the work in [3] by
showing the construction of various QCQP instances satisfying (c).
In addition to
the instance mentioned above,
Figures 1 through 13
illustrate two-dimensional feasible regions satisfying condition (c). We also generate higher-dimensional QCQP instances
by extending the two-dimensional instances.
For the equivalence of a QCQP and its SDP relaxation,
we need to impose some conditions on the quadratic functions
involved in the QCQP. Broadly speaking, there are three types of conditions
that guarantee the equivalence between
QCQPs and their SDP relaxations.
The first type is the convexity of quadratic objective
function and quadratic inequality constraints. QCQPs satisfying this condition are
called as convex quadratic programs.
The second type of conditions concerns the sign pattern of the coefficient matrices in
both of quadratic objective and constraint functions.
A class of QCQPs satisfying this type of conditions was proposed in [17] and
has since been extensively studied in the literature
[4, 7, 13].
The focus of this paper is on the third type which imposes specific conditions only on
quadratic inequality constraints, without any conditions on the objective function.
We note that condition (c) mentioned
above falls under this type.
If the constraints of a QCQP satisfy this type of conditions, then we can arbitrarily choose
any quadratic objective function so that the QCQP is equivalent to its SDP relaxation.
QCQPs satisfying this type of conditions were studied in [1, 2, 3, 9]. Specifically, we focus on
Condition (B)’ below proposed in [3].
We note that Condition (B)’ is translated into condition (c) above.
For the subsequent discussion, we describe the standard form QCQP
as the following:
(1)
Here
The set forms a cone in ; that is, for every
and , it holds that .
It is not convex unless .
We also know that
(the convex hull of ).
We may
assume that is finite in this paper, though
Theorem 1.1 presented below remain valid even when the cardinality of
is infinite.
For every , is
an rank- positive semidefinite
matrix, and can be written as .
Hence we can rewrite (1) as
If we remove rank in the QCQP above (or relax by in QCQP (1)), we obtain an SDP relaxation of QCQP (1):
(2)
In general, holds.
As a special case of QCQP (1), we ocus on the
case where is the diagonal matrix
with the diagonal entries , i.e., .
Then the identity implies either or .
Since
and for every ,
we can fix in this case.
Therefore, we can rewrite QCQP (1) with
as
(3)
To simplify the subsequent discussion, we introduce
the following notation:
for every and .
Using the above notation, we can rewrite QCQP (1), its SDP relaxation (2)
and QCQP (3) as
(4)
(5)
(6)
respectively.
We now consider the following conditions on .
(B)
.
(B)’
for every distinct pair of and .
(C)’
for every .
(D)
There exist such that
for every distinct pair of and .
Conditions (B), (B)’ and (C)’ were introduced in [3, Section 1],
and (D) is a special case of a sufficient condition discussed in [3, Lemma 3.4]
for (B).
Theorem 1.1.
(i)
Let .
Assume that
where denotes the convex hull of
. Then
in QCQP (4) and its SDP relaxation (5).
( is called ROG (Rank-One Generated) in the literature [1, 6]).
(ii)
Condition (B) .
(iii) Condition (B)’ and (C)’ .
(iv) Conditiion (D) Conditions (B) and (B)’.
Proof.
See [9, Theorems 3.1] for (i), [3, Theorems 1.2] for (ii)
and [3, Theorem 1.3] for (iii).
(iv) can be proved easily. To prove (D) (B), let and
. Then we see that
which implies .
To prove (D) (B)’, assume on the contrary that
for some distinct . Then
which is a contradiction.
∎
Remark 1.2.
Assertion (iii) above can be strengthened to
(iii)’ Condition (B)’ .
In fact, we can prove that Condition (C)’ is equivalent to . If
then and . Hence
and
.
Therefore, we will not explicitly refer to Condition (C)’ in the following discussion,
assuming that it is satisfied.
It should be noted that does not
involve and
. By Theorem 1.1, if satisfies on
Condition (B)’ imposed on QCQP (6),
which is a special case of QCQP (4) with
, then holds in QCQP (4) and its SDP relaxation (5)
for every .
In particular,
if we take a positive definite matrix for ,
the feasible regions of both QCQP (4) and its SDP relaxation (5)
are bounded; hence . In this case, we have .
When the feasible region of SDP relaxation (5) is unbounded, however,
may happen even if holds. We will show such a case in Instance 2.7.
Main contribution of the paper
Our first contribution is to present various QCQP instances satisfying Condition (B)’ by introducing a
systematic method to construct such QCQPs.
This work is based on the authors’ previous work [2, 3, 9]. In [3],
Condition (B)’ was proposed as a sufficient condition for the equivalence between QCQP and its SDP relaxation.
A detailed theoretical analysis was also conducted to examine its relationship with several previously proposed sufficient conditions
in [2, 9].
While the paper [3] provided
some examples of QCQPs satisfying Condition (B)’,
it remains unclear what kinds
of problems the entire class of such QCQPs include.
This work seeks to bridge this gap by presenting
various QCQP instances within this class. Specifically, for two-dimensional QCQPs,
we provide instances that fully utilize the geometric properties of Condition (B)’ to make them more intuitive and easier to understand.
Based on these, it is demonstrated that various instances of high-dimensional QCQPs can be constructed freely.
These findings deepen the understanding of Conditions (B)’ proposed in
[3] and are expected to facilitate the broader application of QCQPs that satisfy Conditions (B)’.
The second contribution of this work is the introduction of a numerical method to compute the optimal solution of a QCQP satisfying Condition (B)
from the optimal solution of
its SDP relaxation. The method is based on [16, Lemma 2.2] and its constructive proof, and is
illustrated with numerical
results to demonstrate its effectiveness.
Organization of the paper
In Section 2, we deal with the case . In this case, (6) is a
QCQP in the -dimensional variable vector . After demonstrating how to construct
basic quadratic constraints in , we combine them to generate
instances of satisfying Conditions (B)’ and (D). In Section 3, we discuss how to combine those instances for constructing
higher-dimensional satisfying Condition (D). In Section 4, we show how to compute
an optimal solution of QCQP (4) from
an optimal solution of its SDP relaxation (5)
under Condition (B) and some additional
assumption.
Throughout this section, we assume that and in QCQP (6).
Sections 2.1 and 2.2 describe quadratic constraints for Section 2.3
where instances of satisfying Conditions (B)’ and (D) are presented.
We introduce types of basic quadratic constraints in Section 2.1,
and types of linear transformations for them in Section 2.2.
2.1 Basic quadratic constraints
A disk constraint:
where denotes a parameter. See Figure 1.
Figure 1: The disk constraint : (left), (right).
A hyperbola constraint:
where denotes a parameter. See Figure 2.
Figure 2:
The hyperbola constraint : (left), (right).
A parabola constraint:
where denotes a parameter. See Figure 3.
Figure 3:
The parabola constraint : (left), (right).
A linear constraint:
where denotes a parameter. See Figure 4.
Figure 4: The linear constraint : (left), (right).
2.2 Scaling, rotation and parallel transformation
If we apply a linear transformation defined by , where and denotes a nonsingular matrix,
to the basic quadratic constraints described in Section 2.1,
we obtain general quadratic constraints in .
Under this transformation, the quadratic function
in is transformed into the quadratic function
.
As such a linear transformation, we consider a scaling, a rotation and a parallel
transformation, which are used in Section 2.3 for instances of satisfying
Conditions (B)’ and (D).
We describe matrices for scaling, rotation and parallel transformation.
A scaling matrix:
,
where denotes a parameter vector. The vector
is transformed to .
A rotation matrix: ,
where or denotes a parameter. The vector is transformed to
.
A matrix for parallel transformation:
,
where denotes a parameter vector.
The vector is transformed to .
Figure 5 illustrates the application of scaling, rotation and parallel transformation to
the parabola constraint presented in Section 2.1.
Figure 6 illustrates the application of scaling, rotation and parallel transformation to
the linear constraint given in Section 2.1.
We note that if the order of scaling, rotation and parallel transformation is changed, then
the resulting constraint may differ.
Figure 5: The application of scaling, rotation and linear transformation to
the parabola constraint described in Section 2.1. Here
.
Figure 6:
The application of scaling, rotation and parallel transformation to
the linear constraint
given in Section 2.1.
Here
2.3
Instances of satisfying Conditions (B)’ and (D)
Each in Instances 2.1, 2.2 and 2.3 is
obtained by applying scaling, rotation and/or
parallel transformation to , or for some .
Each in Instances 2.4 and 2.5 is described explicitly without
relying on scaling, rotation or parallel transformation.
Instance 2.6 leads to an extension of satisfying Condition (D)
to a general with in Section 3.
Throughout all instances of ,
is illustrated by the shaded regions,
where the unshaded regions correspond to the feasible regions
of QCQP (6).
Instance 2.7 shows a case where occurs even if satisfies
Condition (B)’.
Instance 2.1.
We define
and
as follows:
(9)
Figure 7 illustrates
, where the unshaded region
corresponds to . We took in the left figure, and in the
right figure. We see from Figure 7 that satisfies Condition (B)’.
Let and . We can verify that
. Therefore,
also satisfies Condition (D).
Figure 7:
Illustration of Instance 2.1. The value of is for the left figure and for the right figure.
Instance 2.2.
Let where denotes the set of integers. We also let and .
Define
See Figure 8. In the left figure, we took , and , where the hyperbola constraint
is shifted to the upper-right direction with the origin to
create .
After rotating by , and then we move the resulting hyperbola toward the
-direction to obtain .
We can make a similar observation on the right figure. From Figure 8, we see that
satisfies Condition (B)’.
We can also verify that
also satisfies Condition (D) with .
Figure 8: Illustration of Instance 2.2. The values for the parameters are and for the left figure, and
and for the right figure.
Instance 2.3.
For every and ,
define
(13)
See Figure 9. In the left figure, is expressed as a scaled parabola constraint such that
and
are obtained by
rotating by and , respectively.
We can make a similar observation on the right figure.
We see from Figure 9 that satisfies Condition (B)’.
Let and . Then
we can verify that
holds.
Therefore, satisfies Condition (D).
Figure 9: Illustration of Instance 2.3. We took for the left figure, and for the right figure
.
Instance 2.4.
In this and next instances, we construct without relying on scaling, rotation and
parallel transformation.
For every and , define
We then see that
It is easy to observe that, for every distinct and every ,
holds.
Hence, if we take a finite subset of satisfying
then satisfies Condition (D).
By Theorem 1.1 (iv), also satisfies Condition (B)’,
as can be verified by Figure 10.
Figure 10: Illustration of Instance 2.4: We took for the left figure,
and
for the right figure.
Instance 2.5.
For every and , define
(14)
Then it follows that
We can easily verify that, for every distinct and every ,
holds.
Therefore, for every finite subset of satisfying
satisfies Condition (D).
By Theorem 1.1 (iv), satisfies Condition (B)’. See Figure 11.
Figure 11: Illustration of Instance 2.5. We took for the left figure,
and
for the right figure.
Instance 2.6.
In this instance, we utilize in (9) (Instance 2.1) and
in (13) (Instance 2.3).
Let as in the left figure of Figure 7, and as in the right figure of Figure 9.
Then each of and consists of matrices in .
As we have stated there, they both satisfy Condition (D);
Letting , and , we consider the convex
combination of and such that
Then
holds.
Therefore satisfies Condition (D). See Figure 12.
If we renumber contained in
before taking the convex combination, a different is obtained.
Figure 12: Illustration of Instance 2.6: We took for the left figure, and
for the right figure.
Instance 2.7.
Define
and let . Then it follows that
(15)
See the right figure of Figure 6. By Theorem 1.1,
the relation
holds between QCQP (6) and its SDP relaxation (5)
with any .
If the objective function is convex in , then (6) becomes a convex QCQP,
so that holds. We now consider a nonconvex case. For example, consider
the case where
Obviously, QCQP (6) attains the minimum at
every boundary points of satisfying .
On the one hand, is a feasible solution of SDP (5) for every , and
as .
Therefore, we obtain in this case.
We note that is represented by two linear inequalities. We now represent
the same by a single quadratic inequality such that
In this case, holds.
In general, SDP relaxation of a QCQP depends on the representation of the QCQP.
In particular, quadratic inequality representation often yields an SDP relaxation that provides a better approximate optimal value (see [5, Theorem 2.1]).
3 An extension to higher-dimensional instances
Several cases of satisfying were provided in
[2, Section 4.1] and [3, Section 6]. In this section,
we show how to construct various ’s that satisfy Condition (D)
by combining multiple ’s .
Recall that we have shown in Instance 2.6 that
a combination of two ’s ,
which have the same cardinality and
satisfy Condition (D), creates a new
satisfying Condition (D). We generalize this technique
to recursively construct a higher-dimensional satisfying Condition (D).
Let . We assume that both
and
satisfy Condition (D).
We note that for every and .
For simplicity of notation, we assume that
.
Thus, our assumption indicates that
Let be an matrix. Then,
we define by
(16)
where denotes the matrix of ’s.
Let . Then,
holds. Therefore, satisfies Condition (D). Furthermore, by replacing by ,
we can continue this procedure recursively.
This recursive procedure is highly flexible as it allows
arbitrary selection of and satisfying Condition (D), as well as
any matrix .
Thus we can create various ’s in by this recursive procedure.
In addition to ’s described in Section 2, which can be employed
for the initial and and also for , we mention some
’s satisfying Condition (D).
(a)
Every satisfies Condition (D). Specifically, every
satisfies Condition (D).
(b)
For every and , let
Then forms a ball with the center and radius . For every
satisfying
(17)
define
Then, we see that
for every distinct . In fact, every leading principal minors of
the matrix above is nonnegative. (In particular, its determinant can be computed with its
Schur complement). Therefore, satisfies Condition (D).
(c)
Suppose that both and satisfy Condition (D), but the number of matrices in
is less than .
To apply (16), we must either discard matrices
from or
introduce ‘dummy’ matrices into
to adjust their cardinalities.
If we take a sufficiently large , then
serves as such a dummy matrix that satisfies Condition (D).
More precisely,
let denote
the minimum eigenvalue of , and . Then satisfies Condition (D).
We now show how to use (b) and (c) for the recursive formula
(16) to construct . Let and .
Then satisfies Condition (D) since it consists of a single matrix in .
For a finite subset of
satisfying (17), let , which satisfies Condition (D).
Suppose that consists of matrices .
By choosing
nonnegative numbers ,
we expand to
, which satisfies Condition (D)
by (c) and
contains the same number of matrices as . Thus we can apply the recursive formula
(16) to construct .
If we take , , , and
,
then corresponds to in Instance 2.4.
We provide two examples of . For simplicity, we assume that , but
the discussion below can be generalized to any in a straightforward manner.
Let .
For the first example, let
Then, through the linear transformation
the quadratic function in
is transformed to
the quadratic function
in .
Hence, if we fix to be , we obtain a convex combination of the two quadratic functions
and
. Thus this case corresponds to Instance 2.6.
Now, we consider the case
In this case, through the linear transformation
the quadratic function
in
is transformed to
the quadratic function
in .
Therefore, if we fix to be ,
we obtain a convex combination of the quadratic function
in and
the quadratic function
in
.
Remark 3.1.
To add a linear equality to QCQP (6), we introduce
, where denotes an matrix,
and . We see that
Hence if and only if . It is known that
if satisfies Condition (B), then so does .
See [2, Section 4.4] for more details.
4 Computing a QCQP optimal solution from an optimal solution of its SDP relaxation
Throughout this section, we assume
(a)
satisfies Condition (B),
(b)
The SDP relaxation (5) of QCQP (4) has an optimal solution
,
(c)
satisfies
the KKT (Karush-Kuhn-Tucker) stationary condition:
there exists a such that
(21)
In (c), corresponds
to an optimal solution of the dual of SDP (5). We note that (c) (b).
If ,
in particular, if as in QCQP (6),
then (b) (c) [8, Theorem 2.1].
By Theorem 1.1, the optimal values of SDP (5) and QCQP (4) coincide, i.e.,
.
We will describe a numerical method for computing an optimal solution
of QCQP (4) from .
Recall that all -dimensional QCQP instances in Section 2.3
satisfy Condition (D) and that a recursive procedure is provided
for constructing higher-dimensional QCQP instances satisfying
Condition (D) in Section 3. Since Condition (D) implies Condition (B) by
Theorem 1.1 (iv), we can apply
the method to those instances.
Let . Then we have either
(i) for some .
(ii) .
We first deal with case (i). In this case, the method is based on the following lemma and its
constructive proof.
Lemma 4.1.
([16, Lemma 2.2], see also [15, Proposition 3])
Let and with
rank. Suppose that . Then, there exists a rank-1
decomposition of such that
and
. If, in particular, , then
.
By Lemma 4.1,
there exists a rank-1 decomposition of such that
and
(i.e., ) .
By assumption (a), .
Since
, there exist a
and a such that .
Let . Since ,
. We also see
that . Hence
is a rank- feasible solution of SDP (5). Furthermore,
we see from and
that
Hence, is a rank-1 optimal solution of SDP (5)
and it is an optimal solution of QCQP (4) with the
same objective value .
We now describe how to compute the rank- decomposition of
such that and
based on the constructive proof of
Lemma 4.1 in [15, 16].
Let be an arbitrary rank- decomposition.
If , then we are done. Otherwise,
there exist and such that , say and .
We consider the following quadratic equation in :
Since , this equation has two distinct roots with
opposite signs. Let be one of the roots. Let
Then, we have
Now replace with and with . Then we still have the
rank- decomposition . If
for some , we continue this procedure recursively till all
are attained.
To compute an optimal solution
of QCQP (4), we can terminate the procedure once we
find such that and .
In this case,
is an optimal solution of QCQP (4).
We now consider case (ii).
In the KKT condition (21), satisfies
and . This implies that
is an optimal solution of a simple SDP with the single equality constraint
:
(22)
The KKT stationary condition for SDP (22) is written as
(26)
for some ,
which serves as a sufficient condition for
to be an optimal solution of SDP (22).
Now we compute a rank- optimal solution of SDP (22) from .
Let . If , then
we have done. So assume . Let be a
rank- decomposition of .
It follows from (21) with that
Since and ,
. Hence the identity above implies
. We also see from
that there exists a such that
. Define . Then
satisfies (26). Thus is a rank- optimal solution of
SDP (22). If then
is a rank- optimal solution of
SDP (5), hence an optimal solution of QCQP (4).
Otherwise, we can take an optimal solution of
SDP (5) such that
as a convex combination of and
, which leads to case (i).
Figure 13: The unshaded region represents the feasible region of QCQP (27).
To illustrate how the method works, we consider the following instance.
Instance 4.2.
We simultaneously describe 6 QCQP problems with quadratic objective functions
over a common quadratic inequality feasible region:
(27)
(28)
where
See Figure 13 for the feasible region of QCQP (27).
Table 1 presents a summary of the numerical results obtained for solving QCQP (27).
The optimal solutions of QCQP (27) can be easily obtained from Figure 13
for ,
as shown in Table 1. The
optimal solutions of the SDP relaxation were computed by SeDuMi [14].
For or , QCQP (27) has a unique optimal solution, and
a rank- solution was obtained by just solving the SDP relaxation.
For ,
QCQP (27) has a unique optimal solution , but
the computed optimal solution of the SDP relaxation has rank- and case (i) occurred.
For , or , the SDP relaxation as well as QCQP (27) have
multiple optimal solutions,
and rank.
For or ,
the rank- decomposition
in case (ii)
led to an optimal solution of QCQP (28) for some
and .
For ,
case (ii) occurred at the optimal solution of the SDP relaxation, but its rank- decomposition
yielded no feasible rank- solution and
the recursive procedure of case (i) was carried out from a convex combination of for some and .
QCQP
SDP, : Computed Optimal Solution
Opt.Sol.
Opt.Val.
Opt.Val
Rank
Case
1
0
1
5.00
1.00
1.00
(ii)
2
4
1
6.00
3.00
(i)
3
-2
2
6.00
5.26
(i)
4
0
3
2.00
3.99
2.28
(ii)
5
0
2
1.68
1.42
(ii)
6
0
2
3.45
2.55
7.55
(ii) (i)
Table 1:
.
.
.
5 Concluding remarks
If satisfies , then QCQP (4) is equivalent to
its SDP relaxation (5) whose optimal value and solution can be easily computed.
In Section 3, we have shown a recursive procedure for constructing
various ’s satisfying Condition (D).
The class of QCQPs with such ’s appears to be quite broad,
at least in theory. It should be noted
that Condition (D) is merely sufficient for .
For example, we could relax Condition (D) to
Condition (D)’:
If and then
for some nonzero ([1, Proposition 1]).
There is still a gap, however, between Condition (D)’ and the condition
([1, Theorem 1] and [3, Example 6.1]).
See [3, Section 3]
where various sufficient conditions for
and
their relationships are discussed, under moderate assumptions including the
Slater constraint qualification that
contains a positive definite matrix.
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