Local Convergence of Sequential Convex Programming for Nonconvex Optimization

Dinh, Quoc Tran; Diehl, Moritz

doi:10.1007/978-3-642-12598-0_9

Cited by 146 publications

(156 citation statements)

References 9 publications

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“…In the same spirit as the MMSE design for time multiplexing (9), the MMSE design of time and color multiplexing codes can be carried out by minimizing the MMSE (19) under the constraint that (20) The correlation matrix of the demultiplexed images can be obtained during the camera calibration phase; see Section VI-B. Concretely, we aim to solve the optimization problem (21) By comparison to straightforward time and color multiplexing, i.e., and , the gain of optimal time and color multiplexing is defined as (22) where is the noise covariance given by (17) with . The objective function of problem (19) is highly non-convex.…”

Section: The Mmse Design Of Time and Color Multiplexing Codesmentioning

confidence: 99%

“…Some approximation and decomposition of problem (19) should be sought, with the hope that the solution found is less-sensitive to initial condition and that the subproblems can be convex and be easy to solve. In the following, we reformulate and approximate problem (21), shown at the bottom of the page, in such a way that the problem can be easily handled by sequential convex programming (SCP) and a 1-dimensional exhaustive search over a set of finite samples, where each subproblem involved in the SCP step is casted as a semi-definite programming (SDP) problem that can be readily solved by any convex optimization solver. We name the proposed method the Illumination Multiplexing Codes (IMC) algorithm.…”

Section: The Mmse Design Of Time and Color Multiplexing Codesmentioning

confidence: 99%

“…We apply a 1-dimensional exhaustive search over finite samples of , say , where we set given by (14) as an upper bound to account for multiplexing saturation problems. After we obtain the optimal value for , a solution of and the time and color multiplexing matrix to problem (24) can be found by (35) (36) Finally, we summarize the IMC algorithm for problem (21) in Algorithm 2. The IMC algorithm can be readily applied to the MMSE design of time multiplexing codes only.…”

Section: ) Exhaustive Search Over Various 'Smentioning

confidence: 99%

“…Unlike conventional sequential SDP [20] which applies Newton's method to solve the Karush-Kuhn-Tucker conditions of (31), the proposed Algorithm 1 employs Newton's method on the non-convex constraint only. The local convergence for this type of algorithm has been analyzed in [21], whose results relate to the following nonconvex optimization problem,…”

Section: ) Convergence Analysis Of Algorithmmentioning

confidence: 99%

“…The authors of [21] studied the iterative method (38) for and showed that, under some contraction properties, if is a stationary point of (37) then within some neighborhood of the sequence generated by (38) converges linearly to .…”

Section: ) Convergence Analysis Of Algorithmmentioning

confidence: 99%

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MMSE Design of Time and Color Multiplexing Codes

Chan

Jia

Wycoff

et al. 2014

IEEE Trans. Signal Process.

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Abstract-Illumination multiplexing has proven itself to be a valuable tool in image quality improvement for many computer vision and graphics applications, provided that its limitations on photon noise and saturation are properly tackled. Currently, multiplexing codes are constructed according to the maximum signal-to-noise ratio (SNR), and they are primarily employed for time multiplexing, a technique that requires the number of measurements to be equal to the number of illumination sources . In this work, we propose an illumination multiplexing method based on minimum mean square errors (MMSEs) for the more general setting of both time and color multiplexing performed together, with , where is the number of color channels. Under the umbrella of the proposed MMSE formulation, the conventional maximum SNR approach can be thought of as a special case of the MMSE design. The formulated MMSE problem is a difficult non-convex problem, but it can be approximated by sequential semi-definite (convex) programming and a 1-D exhaustive search. The proposed formulation and algorithm can be readily specialized to max-SNR and/or time multiplexing designs, thereby giving the optimized codes a much broader scope of application. Computer simulations show that the conventional max-SNR design is suboptimal to the proposed MMSE design, though both see significant quality improvements as increases. Experiments also demonstrate the effectiveness and superiority of the proposed method in illuminating various objects.Index Terms-Optical multiplexing, maximum signal-to-noise ratio (SNR), minimum mean square error (MMSE), sequential convex approximation, time and color.

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Section: The Mmse Design Of Time and Color Multiplexing Codesmentioning

confidence: 99%

Section: The Mmse Design Of Time and Color Multiplexing Codesmentioning

confidence: 99%

Section: ) Exhaustive Search Over Various 'Smentioning

confidence: 99%

Section: ) Convergence Analysis Of Algorithmmentioning

confidence: 99%

Section: ) Convergence Analysis Of Algorithmmentioning

confidence: 99%

See 3 more Smart Citations

MMSE Design of Time and Color Multiplexing Codes

Chan

Jia

Wycoff

et al. 2014

IEEE Trans. Signal Process.

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Operating Protocols and Beamforming Design for STARS

Liu,

Mu,

et al. 2024

Wireless Networks

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Complexity of linearized quadratic penalty for optimization with nonlinear equality constraints

Bourkhissi,

Necoara

2024

J Glob Optim

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In this paper we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints, are locally smooth. For solving this problem, we propose a linearized quadratic penalty method, i.e., we linearize the objective function and the functional constraints in the penalty formulation at the current iterate and add a quadratic regularization, thus yielding a subproblem that is easy to solve, and whose solution is the next iterate. Under a new adaptive regularization parameter choice, we provide convergence guarantees for the iterates of this method to an $$\epsilon $$ ϵ first-order optimal solution in $${\mathcal {O}}({\epsilon ^{-2.5}})$$ O ( ϵ - 2.5 ) iterations. Finally, we show that when the problem data satisfy Kurdyka–Lojasiewicz property, e.g., are semialgebraic, the whole sequence generated by the proposed algorithm converges and we derive improved local convergence rates depending on the KL parameter. We validate the theory and the performance of the proposed algorithm by numerically comparing it with some existing methods from the literature.

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Local Convergence of Sequential Convex Programming for Nonconvex Optimization

Cited by 146 publications

References 9 publications

MMSE Design of Time and Color Multiplexing Codes

MMSE Design of Time and Color Multiplexing Codes

Operating Protocols and Beamforming Design for STARS

Complexity of linearized quadratic penalty for optimization with nonlinear equality constraints

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