Constant-Modulus Multi-Order CQP

• CMCQP is an optimization framework featuring complex constant-modulus variables and multi-order quadratic objectives that model radar waveform, beamforming, and other signal processing metrics.
• The approach employs phase-only reformulation alongside methods such as steepest descent, successive convexification, consensus-ADMM, and enhanced semidefinite relaxations to mitigate nonconvexity.
• Empirical results indicate that CMCQP techniques achieve competitive SINR and beampattern performance with improved computational efficiency in applications like MIMO radar and robust beamforming.

Constant-Modulus Multi-Order Complex Quadratic Programming (CMCQP) denotes a class of optimization problems in which the decision variable is complex and every entry has fixed modulus, while the objective is a multi-order complex quadratic form. In a generalized formulation, the variable is $\mathbf{x}\in\mathbb{C}^N$ with $|x_n|=c$, and one minimizes or maximizes either $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$ or a sum $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$. The objective function typically relates to metrics such as signal-to-noise ratio, Cramér-Rao bound, integrated sidelobe level, and related waveform or beamforming criteria, while the constraints normally correspond to requirements on similarity to desired aspects, peak-to-average-power ratio, or constant-modulus property in practical scenarios. In general, CMCQP is non-convex and difficult to solve; in application-specific settings such as MIMO radar waveform design with signal-dependent clutter and additive white Gaussian noise, the objective becomes effectively quartic or rational in the waveform, and the resulting design problem is described as non-convex and NP-hard (Shi et al., 27 Aug 2025, Ren et al., 2018).

1. Formal problem class

A canonical generalized formulation is

$$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$

or the corresponding maximization problem, with $q>0$ and $\mathbf{A}\in\mathbb{C}^{N\times N}$. A broader version uses sums of terms,

$$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$

under the same constant-modulus constraints. This formulation is presented as a unifying abstraction for a range of signal processing problems (Shi et al., 27 Aug 2025).

Two representative cases are emphasized. In CaseA1, $\mathbf{A}$ is arbitrary and $q=2$, so the objective is $|x_n|=c$0; the paper associates this case with auto/cross-correlation shaping and ISL/WISL, where quartic objectives arise naturally. In CaseA2, $|x_n|=c$1 and $|x_n|=c$2, so the modulus can be removed because $|x_n|=c$3 is real; this directly models metrics such as SNR/SINR, CRB surrogates, beamspace energy, AF shaping, and PAPR-related quadratic forms (Shi et al., 27 Aug 2025).

A concrete radar instantiation appears in MIMO waveform design. After elimination of the unconstrained receive filter, the transmit-only problem becomes

$|x_n|=c$4

where $|x_n|=c$5 and $|x_n|=c$6 depends quadratically on $|x_n|=c$7 through the signal-dependent clutter covariance (Ren et al., 2018).

2. Multi-order structure and feasible-set geometry

The phrase “multi-order” refers to more than one mechanism. In the generalized formulation, it can mean that the complex quadratic form is raised to an arbitrary positive power $|x_n|=c$8, or that multiple such terms are aggregated in a sum. In application-driven formulations, it can also refer to higher-than-quadratic dependence introduced indirectly through embedded quadratic structure. In the MIMO radar problem, $|x_n|=c$9 appears inside a matrix inverse, so the objective $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$0 is not purely quadratic and is described as effectively quartic or rational in $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$1 (Ren et al., 2018).

Constant-modulus constraints admit a phase-only reparameterization. With $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$2, one may write

$\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$3

or, for $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$4,

$\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$5

This converts the constrained optimization into an unconstrained optimization in the phase vector $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$6. For the generalized form,

$\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$7

The same phase-only viewpoint underlies similarity-constrained radar waveform design, where the infinity-norm similarity constraint becomes the per-entry phase-interval constraint $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$8 under constant modulus (Shi et al., 27 Aug 2025, Ren et al., 2018).

The feasible set is therefore nonconvex even before one accounts for the objective. In the radar construction, when $\big|\mathbf{x}^H\mathbf{A}\mathbf{x}\big|^q$9, the constant-modulus arc and similarity interval are relaxed to the convex hull formed by a disk constraint $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$0 intersected with a halfspace $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$1; when $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$2, the linear inequality reverses sign. In semidefinite relaxations with pairwise phase-difference constraints, the convex hull of admissible off-diagonal entries is described explicitly: for continuous intervals, $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$3 is characterized by one linear inequality and $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$4, while for discrete phase sets it becomes a polyhedral set given by finitely many linear inequalities (Ren et al., 2018, Xu et al., 2023).

A related multi-order constant-modulus structure appears in constrained constant modulus beamforming. There the classical CM cost is

$$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$5

which is quartic in $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$6. The paper then uses an iterative second-order approximation to obtain a quadratic surrogate in $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$7, thereby moving from a quartic constant-modulus objective to a tractable quadratic program with robust SOC constraints (Landau et al., 2013).

3. Principal algorithmic approaches

One major line of work reformulates CMCQP as unconstrained optimization over phases and then applies steepest descent or ascent. For Hermitian PSD $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$8 with $$\sum_k \alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}$$9, the phase gradient is

$$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$0

while for the arbitrary-$$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$1, $$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$2 case, the gradient involves both $$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$3 and $$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$4. The step-size search is converted to a polynomial form via a third-order Taylor expansion,

$$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$5

and the positive critical point of the cubic derivative is used as a closed-form step size whenever valid. The resulting Min/Max-CMCQP$$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$6 and Min/Max-CMCQP$$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$7 methods have per-iteration complexity $$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$8, with constants about $$\min_{\mathbf{x}\in \mathbb{C}^N}\quad \big|\mathbf{x}^{H}\mathbf{A}\mathbf{x}\big|^q \quad\text{s.t.}\quad |x_n|=c,\; n=1,\dots,N,$$9 for CaseA1 and $q>0$0 for CaseA2, and the paper also allows optional SQUAREM acceleration (Shi et al., 27 Aug 2025).

A second line of work uses successive convexification. In the MIMO radar formulation, $q>0$1 is frozen at the current iterate to obtain a quadratic surrogate $q>0$2. After choosing $q>0$3 and setting $q>0$4, the subproblem is relaxed to a convex QCQP and solved by Accelerated Gradient Projection (AGP). The AGP update is FISTA-like,

$q>0$5

followed by a tailored per-entry projection onto the relaxed feasible set and a final phase normalization

$q>0$6

Its per-iteration complexity is dominated by $q>0$7, whereas the interior-point baseline is stated as $q>0$8 (Ren et al., 2018).

Consensus-ADMM provides a different decomposition strategy for QCQP-type instances. The problem is written in consensus form with local copies $q>0$9 for each constraint set $\mathbf{A}\in\mathbb{C}^{N\times N}$0, including the constant-modulus set $\mathbf{A}\in\mathbb{C}^{N\times N}$1. The $\mathbf{A}\in\mathbb{C}^{N\times N}$2-update solves the Hermitian linear system

$\mathbf{A}\in\mathbb{C}^{N\times N}$3

while the constant-modulus $\mathbf{A}\in\mathbb{C}^{N\times N}$4-update is the exact element-wise projection

$\mathbf{A}\in\mathbb{C}^{N\times N}$5

Additional quadratic constraints are handled as QCQP-1 projections through scalar secular equations. The paper emphasizes scalability, memory-efficient implementation, parallel or distributed execution, and smart initialization, while noting only a weak convergence result for the nonconvex setting (Huang et al., 2016).

A further framework, Extreme Point Pursuit (EXPP), replaces the nonconvex constant-modulus set $\mathbf{A}\in\mathbb{C}^{N\times N}$6 by its convex hull $\mathbf{A}\in\mathbb{C}^{N\times N}$7 and solves

$\mathbf{A}\in\mathbb{C}^{N\times N}$8

Projected gradient then takes the form

$\mathbf{A}\in\mathbb{C}^{N\times N}$9

For quadratic $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$0 with Hermitian $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$1, the paper states $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$2, and for $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$3 the penalized problem is globally equivalent to the original CM problem because the objective becomes strictly concave on the convex hull and the minimizer is driven to an extreme point (Liu et al., 2024).

4. Relaxations and global optimization

Semidefinite relaxation is a central tool for CMCQP and adjacent complex QCQP formulations. In lifted form, $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$4 transforms each quadratic term into a linear form $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$5, and constant modulus becomes $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$6. Conventional SDRs drop the rank-one constraint and often also relax explicit phase structure, but this can be loose when phases are tightly constrained (Xu et al., 2023).

Enhanced SDRs strengthen the lifted model by using polar-coordinate geometry. One family introduces an auxiliary real matrix $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$7 and pairwise convex-hull constraints linking $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$8. The first enhanced relaxation, ECSDP1, enforces $$f(\mathbf{x})=\sum_{k}\alpha_k\big|\mathbf{x}^H\mathbf{A}_k\mathbf{x}\big|^{q_k}, \quad \alpha_k\in\mathbb{R}_+,\; q_k>0,$$9 and $\mathbf{A}$0; the second, ECSDP, adds $\mathbf{A}$1. The paper proves a pairwise exactness property: the projection of the feasible set onto $\mathbf{A}$2 equals $\mathbf{A}$3 for each constrained pair. In the unit-modulus case $\mathbf{A}$4, these conditions force $\mathbf{A}$5, $\mathbf{A}$6, and $\mathbf{A}$7, so the remaining freedom lies in the allowed phase differences. The same work states that the new relaxations can be strictly tighter than previous ones and can be applied to more general cases than earlier SDRs designed only for special cases (Xu et al., 2023).

A closely related enhanced SDR, ECSDR, is built directly from single-variable modulus and angle envelopes. For the baseline complex quadratic program,

$\mathbf{A}$8

the constant-modulus ECSDR reads

$\mathbf{A}$9

This relaxation is then embedded in a branch-and-bound algorithm, ECSDR-BB, which uses the ECSDR optimum as a lower bound, obtains an upper bound by scaling $q=2$0 to the nearest feasible angle, branches on the most violated angle or modulus component, and returns an $q=2$1-optimal global solution for any $q=2$2 (Lu et al., 2019).

These approaches differ in purpose. Enhanced SDRs provide tighter bounds and improved suboptimal solutions; branch-and-bound built on those bounds aims at global optimality within a prescribed tolerance. By contrast, surrogate-QCQP, phase-gradient, ADMM, and penalized projected-gradient methods aim at scalable local or stationary solutions. The distinction is methodological rather than terminological: all are used on problems with constant-modulus feasible sets and complex quadratic or higher-order structure (Lu et al., 2019, Xu et al., 2023).

5. Representative applications and empirical behavior

CMCQP is presented as a unifying model for waveform/code correlation shaping, SNR/SINR optimization, CRB surrogates, AF shaping, beamspace energy design, and PAPR-related formulations. The ISL/WISL case is written as a sum of terms $q=2$3 under unimodular constraints; optimum-detection SNR/SINR becomes $q=2$4 with $q=2$5; a MIMO-OFDM PAPR surrogate is expressed as $q=2$6 under $q=2$7. The same generalized picture includes CRB and AF shaping after quadratic reformulation through Fisher information or beamspace matrices (Shi et al., 27 Aug 2025).

In the MIMO radar waveform design study, the simulations use $q=2$8, $q=2$9, $|x_n|=c$00, a reference waveform from an orthogonal chirp matrix, target angle $|x_n|=c$01, clutter angles $|x_n|=c$02, $|x_n|=c$03, $|x_n|=c$04, target power $|x_n|=c$05 dB, clutter powers $|x_n|=c$06 dB, and noise variance $|x_n|=c$07 dB. For $|x_n|=c$08, AGP and IPM yield very close SINR, while IPM exhibits better beampattern suppression; for $|x_n|=c$09, AGP’s beampattern outperforms IPM and the SINR difference is less than $|x_n|=c$10 dB; as $|x_n|=c$11 increases with $|x_n|=c$12, AGP achieves notably superior SINR compared to IPM while maintaining much lower computational time (Ren et al., 2018).

The generalized phase-based algorithms are accompanied by detailed numerical comparisons. With $|x_n|=c$13, the cubic step-size accuracy relative to exact line search reaches nearly $|x_n|=c$14 after about one iteration for CaseA1 minimization and about four iterations for CaseA1 maximization; for CaseA2 it reaches nearly $|x_n|=c$15 by about three to six iterations. For CaseA2 minimization with $|x_n|=c$16, after $|x_n|=c$17 iterations Min-CMCQP II attains $|x_n|=c$18 dB normalized objective, compared with $|x_n|=c$19 dB for IA-CPC, $|x_n|=c$20 dB for ADPM, and $|x_n|=c$21 dB for PM-L. For CaseA2 maximization with $|x_n|=c$22, after $|x_n|=c$23 iterations Max-CMCQP II reaches $|x_n|=c$24 dB, compared with $|x_n|=c$25 dB for Newton and $|x_n|=c$26 dB for RCG, while the maximum achievable value is $|x_n|=c$27 dB. In WISL minimization with $|x_n|=c$28 and $|x_n|=c$29, the proposed method reports $|x_n|=c$30 s and $|x_n|=c$31 iterations, versus $|x_n|=c$32 s and $|x_n|=c$33 iterations for WISLNew, $|x_n|=c$34 s and $|x_n|=c$35 iterations for MM-WeCorr, and $|x_n|=c$36 s and $|x_n|=c$37 iterations for WeCAN (Shi et al., 27 Aug 2025).

Enhanced relaxations also show measurable gains. In phase-quantized waveform design with $|x_n|=c$38, $|x_n|=c$39, and $|x_n|=c$40, ECSDP significantly tightens upper bounds over classical SDR, and lower bounds from ECSDP-driven rounding are uniformly better. In discrete transmit beamforming, ECSDP reduces the relaxation gap by about $|x_n|=c$41–$|x_n|=c$42 on average compared with classical SDR. In the ECSDR-BB study, the enhanced relaxation closes a dominant fraction of the conventional SDR gap in MIMO detection and gives a globally convergent algorithm that significantly outperforms sphere decoding in hard cases such as $|x_n|=c$43, $|x_n|=c$44-PSK, and SNR $|x_n|=c$45 dB, where the reported averages are about $|x_n|=c$46 s for ECSDR-BB versus about $|x_n|=c$47 s for sphere decoding (Xu et al., 2023, Lu et al., 2019).

Robust adaptive beamforming provides another constant-modulus multi-order example. The worst-case constrained constant modulus design replaces the CMV quadratic objective by a CCM quadratic surrogate under robust SOC constraints. The reported simulations show that WC-CCM consistently achieves higher SINR than WC-CMV and loaded-SMI under steering-vector mismatch, and that the low-complexity Robust-CCM-MCG algorithm retains quadratic complexity $|x_n|=c$48, more than an order of magnitude lower than interior-point SOC solvers stated as $|x_n|=c$49 (Landau et al., 2013).

6. Limitations, misconceptions, and methodological boundaries

CMCQP is not identical to an ordinary complex QCQP. In some instances the objective is genuinely quadratic after phase-only reformulation or matrix aggregation, but in others the higher-order structure is essential: the generalized model includes powers $|x_n|=c$50 and sums thereof, the MIMO radar SINR objective depends on $|x_n|=c$51 inside a matrix inverse, and the classical constrained constant modulus beamforming cost is quartic before surrogate construction. Treating all such cases as standard QCQPs obscures the role of surrogate design, lifting, or successive approximation (Shi et al., 27 Aug 2025, Ren et al., 2018, Landau et al., 2013).

The main algorithmic limitation remains nonconvexity. The generalized CMCQP paper states explicitly that nonconvexity implies potential convergence to local stationary points and that performance can be sensitive to initialization. Its cubic step-size approximation assumes sufficiently small steps relative to the gradient norm, so early iterations may require safeguarded fallback rules. In the radar AGP construction, the methodology depends on freezing the multi-order term and on separable convex relaxations; if additional non-separable constraints such as strict autocorrelation sidelobe bounds or spectral masks coupling samples and antennas are added, the simple per-entry projection may no longer be available (Shi et al., 27 Aug 2025, Ren et al., 2018).

Relaxation-based methods have complementary weaknesses. Enhanced SDRs provide tighter bounds than classical formulations, but solving an SDP with $|x_n|=c$52 typically scales as $|x_n|=c$53 per interior-point iteration with memory $|x_n|=c$54, and ECSDP adds a second matrix $|x_n|=c$55 plus per-edge linear or SOCP constraints whose overhead depends on $|x_n|=c$56. Consensus-ADMM improves scalability for many QCQP instances, but the nonconvex setting does not come with global convergence guarantees; the cited theorem gives only that any limit point is KKT under conditions such as vanishing consensus gaps and successive differences (Xu et al., 2023, Huang et al., 2016).

Penalty-based convex-hull approaches are likewise conditional. EXPP gives exact penalization results when $|x_n|=c$57 has $|x_n|=c$58-Lipschitz gradient and $|x_n|=c$59, or under appropriate error-bound conditions for nonsmooth $|x_n|=c$60. The same source also states that a universal exact error bound based solely on $|x_n|=c$61 does not exist for arbitrary CM sets, although controlled inexactness of order $|x_n|=c$62 is available. A plausible implication is that convex-hull penalization is best viewed not as a universal replacement for manifold, SDR, or branch-and-bound methods, but as one member of a broader toolkit whose suitability depends on whether the dominant difficulty lies in phase geometry, higher-order objective structure, or global optimality requirements (Liu et al., 2024).

Across these formulations, CMCQP functions less as a single algorithm than as a mathematical umbrella. Its central content is the conjunction of complex quadratic or higher-order structure with constant-modulus geometry; the diversity of solution methods reflects the diversity of that conjunction. Phase-gradient methods, successive QCQP refinement, consensus splitting, enhanced semidefinite relaxation, branch-and-bound, SOC reformulation, and convex-hull penalization each formalize a different compromise among exactness, scalability, and structural fidelity (Shi et al., 27 Aug 2025, Lu et al., 2019).