Penalized semidefinite programming for quadratically-constrained quadratic optimization


In this paper, we give a new penalized semidefinite programming approach for non-convex quadratically-constrained quadratic programs (QCQPs). We incorporate penalty terms into the objective of convex relaxations in order to retrieve feasible and near-optimal solutions for non-convex QCQPs. We introduce a generalized linear independence constraint qualification (GLICQ) criterion and prove that any GLICQ regular point that is sufficiently close to the feasible set can be used to construct an appropriate penalty term and recover a feasible solution. Inspired by these results, we develop a heuristic sequential procedure that preserves feasibility and aims to improve the objective value at each iteration. Numerical experiments on large-scale system identification problems as well as benchmark instances from the library of quadratic programming demonstrate the ability of the proposed penalized semidefinite programs in finding near-optimal solutions for non-convex QCQP.

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The authors are grateful to GAMS Development Corporation for providing them with unrestricted access to a full set of solvers throughout the project.

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Correspondence to Alper Atamtürk.

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This work is in part supported by the NSF Award 1809454. Javad Lavaei is supported by an AFOSR YIP Award and ONR N000141712933. Alper Atamtürk is supported, in part, by Grant FA9550-10-1-0168 from the Office of the Assistant Secretary of Defense for Research & Engineering, NSF Award 1807260, DOE ARPA-E Grant 260801540061, and DOD ONR Grant 12951270.


Application to polynomial optimization

In this section, we show that the proposed penalized SDP approach can be used for polynomial optimization as well. A polynomial optimization problem is formulated as

$$\begin{aligned} \underset{\varvec{x}\in \mathbb {R}^n}{\text {minimize}} \ \&u_0(\varvec{x})&\end{aligned}$$
$$\begin{aligned} \ \ \ \text {s.t.} \ \ \&u_k(\varvec{x})\le 0, \quad \quad k\in \mathcal {I}\end{aligned}$$
$$\begin{aligned}&u_k(\varvec{x})=0, \quad \quad k\in \mathcal {E}, \end{aligned}$$

for every \(k\in \{0\}\cup \mathcal {I}\cup \mathcal {E}\), where each function \(u_k:\mathbb {R}^n\rightarrow \mathbb {R}\) is a polynomial of arbitrary degree. Problem (48a)–(48c) can be reformulated as a QCQP of the form:


where \(\varvec{y}\in \mathbb {R}^{|\mathcal {O}|}\) is an auxiliary variable, and \(v_1,\ldots ,v_{|\mathcal {O}|}\) and \(w_0,w_1,\ldots ,w_{|\{0\}\cup \mathcal {I}\cup \mathcal {E}|}\) are quadratic functions with the following properties:

  • For every \(\varvec{x}\in \mathbb {R}^n\), the function \(\varvec{v}(\varvec{x},\cdot ):\mathbb {R}^{|\mathcal {O}|}\rightarrow \mathbb {R}^{|\mathcal {O}|}\) is invertible,

  • If \(\varvec{v}(\varvec{x},\varvec{y})=\varvec{0}_n\), then \(w_k(\varvec{x},\varvec{y}) = u_k(\varvec{x})\) for every \(k\in \{0\}\cup \mathcal {I}\cup \mathcal {E}\).

Based on the above properties, there is a one-to-one correspondence between the feasible sets of (48a)–(48c) and (49a)–(49d). Moreover, a feasible point is an optimal solution to the QCQP (49a)–(49d) if and only if is an optimal solution to the polynomial optimization problem (48a)–(48c).

Theorem 3

[51] Suppose that \(\{u_k\}_{k\in \{0\}\cup \mathcal {I}\cup \mathcal {E}}\) are polynomials of degree at most d, consisting of m monomials in total. There exists a QCQP reformulation of the polynomial optimization (48a)–(48c) in the form of (49a)–(49d), where \({|\mathcal {O}|}\le m n \left( \lfloor \log _2(d)\rfloor +1\right) \).

The next proposition shows that the LICQ regularity of a point \(\hat{\varvec{x}}\in \mathbb {R}^n\) is inherited by the corresponding point \((\hat{\varvec{x}},\hat{\varvec{y}})\in \mathbb {R}^n\times \mathbb {R}^o\) of the QCQP reformulation (49a)–(49d).

Proposition 1

Consider a pair of vectors \(\hat{\varvec{x}}\in \mathbb {R}^n\) and \(\hat{\varvec{y}}\in \mathbb {R}^{|\mathcal {O}|}\) satisfying \(\varvec{v}(\hat{\varvec{x}},\hat{\varvec{y}})=\varvec{0}_n\). The following two statements are equivalent:

  1. 1.

    \(\hat{\varvec{x}}\) is feasible and satisfies the LICQ condition for the polynomial optimization problem (48a)–(48b).

  2. 2.

    \((\hat{\varvec{x}},\hat{\varvec{y}})\) is feasible and satisfies the LICQ condition for the QCQP (49a)–(49d).


From \(\varvec{u}(\hat{\varvec{x}})=\varvec{w}(\hat{\varvec{x}},\hat{\varvec{y}})\) and the invertiblity assumption for \(\varvec{v}(\hat{\varvec{x}},\cdot )\), we have

$$\begin{aligned} \frac{\partial \varvec{u}(\hat{\varvec{x}})}{\partial \varvec{x}}&= \begin{bmatrix} \frac{\partial \varvec{w}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{x}}&\frac{\partial \varvec{w}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{y}} \end{bmatrix} \begin{bmatrix} \varvec{I}&- \left( \frac{\partial \varvec{v}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{y}} \right) ^{ -1} \frac{\partial \varvec{v}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{x}} \end{bmatrix}^{ \top }\nonumber \\&= \frac{\partial \varvec{w}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{x}} - \frac{\partial \varvec{w}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{y}} \left( \frac{\partial \varvec{v}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{y}} \right) ^{ -1} \frac{\partial \varvec{v}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{x}}. \end{aligned}$$

Therefore, \(\mathcal {J}_{\mathrm {PO}}(\hat{\varvec{x}})=\frac{\partial \varvec{u}(\hat{\varvec{x}})}{\partial \varvec{x}}\) is equal to the Schur complement of

$$\begin{aligned} \mathcal {J}_{\mathrm {QCQP}}(\hat{\varvec{x}},\hat{\varvec{y}})= \begin{bmatrix} \frac{\partial \varvec{w}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{x}} &{} \frac{\partial \varvec{w}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{y}}\\ \frac{\partial \varvec{v}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{x}} &{} \frac{\partial \varvec{v}(\hat{\varvec{x}},\hat{\varvec{y}})}{\partial \varvec{y}}\\ \end{bmatrix}, \end{aligned}$$

which is the Jacobian matrix of the QCQP (49a)–(49d) at the point \((\hat{\varvec{x}},\hat{\varvec{y}})\). As a result, the matrix \(\mathcal {J}_{\mathrm {PO}}(\hat{\varvec{x}})\) is singular if and only if \(\mathcal {J}_{\mathrm {QCQP}}(\hat{\varvec{x}},\hat{\varvec{y}})\) is singular. \(\square \)

Reformulation-linearization technique

This appendix covers the reformulation-linearization technique (RLT) of Sherali and Adams [72] as an approach to strengthen convex relaxations of the form (4a)–(4d) in the presence of affine constraints. Define \(\mathcal {L}\) as the set of affine constrains in the QCQP (1a)–(1c), i.e., \(\mathcal {L}\triangleq \{k\in \mathcal {I}\cup \mathcal {E}\;|\;\varvec{A}_k=\varvec{0}_{n\times n}\}\). Define also


where \(\varvec{B}\triangleq [\varvec{b}_1,\ldots ,\varvec{b}_{|\mathcal {I}\cap \mathcal {E}|}]^{\top }\) and \(\varvec{c}\triangleq [c_1,\ldots ,c_{|\mathcal {I}\cap \mathcal {E}|}]^{\top }\). Every \(\varvec{x}\in \mathcal {F}\) satisfies

$$\begin{aligned} \varvec{H}\varvec{x} + \varvec{h} \le 0, \end{aligned}$$

and, as a result, all elements of the matrix

$$\begin{aligned} \varvec{H}\varvec{x}\varvec{x}^{ \top } \varvec{H}^{\top }+ \varvec{h}\varvec{x}^{ \top } \varvec{H}^{ \top }+ \varvec{H}\varvec{x}\varvec{h}^{ \top }+ \varvec{h}\varvec{h}^{ \top } \end{aligned}$$

are non-negative if \(\varvec{x}\) is feasible. Hence, the inequality

$$\begin{aligned} \varvec{e}^{\top }_i\varvec{V}(\varvec{x},\varvec{x}\varvec{x}^{\top })\varvec{e}_j\ge 0 \end{aligned}$$

holds true for every \(\varvec{x}\in \mathcal {F}\) and \((i,j)\in \mathcal {H}\times \mathcal {H}\), where \(\varvec{V}:\mathbb {R}^n\times \mathbb {S}_n\rightarrow \mathbb {S}_{|\mathcal {H}|}\) is defined as

$$\begin{aligned} \varvec{V}(\varvec{x},\varvec{X})\triangleq \varvec{H}\varvec{X}\varvec{H}^{\top }+ \varvec{h}\varvec{x}^{\top } \varvec{H}^{\top }+ \varvec{H}\varvec{x}\varvec{h}^{ \top }+ \varvec{h}\varvec{h}^{\top }, \end{aligned}$$

\(\mathcal {H}\triangleq \{1,\ldots ,|\mathcal {L}\cap \mathcal {I}|+2|\mathcal {L}\cap \mathcal {E}|\}\), and \(\varvec{e}_1,\ldots ,\varvec{e}_{|\mathcal {H}|}\) denote the standard bases in \(\mathbb {R}^{|\mathcal {H}|}\).

This leads to a strengthened relaxation of QCQP (1a)–(1c):

$$\begin{aligned} \underset{\begin{subarray}{l} \varvec{x}\in \mathbb {R}^n, \varvec{X}\in \mathbb {S}_n\end{subarray}}{\text {minimize}} \ \&\bar{q}_0(\varvec{x},\varvec{X}) \end{aligned}$$
$$\begin{aligned} \ \ \ \ \ \ \text {s.t.} \ \ \ \&\bar{q}_k(\varvec{x},\varvec{X}) \le 0,\qquad k\in \mathcal {I} \end{aligned}$$
$$\begin{aligned}&\bar{q}_k(\varvec{x},\varvec{X}) = 0,\qquad k\in \mathcal {E} \end{aligned}$$
$$\begin{aligned}&\varvec{X}-\varvec{x}\varvec{x}^\top \succeq _{\mathcal {C}_r}0 \end{aligned}$$
$$\begin{aligned}&\varvec{e}^{\top }_i\varvec{V}(\varvec{x},\varvec{X})\varvec{e}_j\ge 0, \qquad \ (i,j)\in \mathcal {V} \end{aligned}$$

where \(\mathcal {V}\subseteq \mathcal {H}\times \mathcal {H}\) is a selection of RLT inequalities.

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Madani, R., Kheirandishfard, M., Lavaei, J. et al. Penalized semidefinite programming for quadratically-constrained quadratic optimization. J Glob Optim (2020).

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  • Semidefinite programming
  • Non-convex optimization
  • Non-linear programming
  • Convex relaxation

Mathematics Subject Classification

  • 65K05
  • 90-08
  • 90C26
  • 90C22