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Three-dimensional (3-D) imaging sonar systems require large planar arrays, which incur hardware costs. In contrast, a cross array consisting of two perpendicular linear arrays can also support 3-D imaging while dramatically reducing the number of sensors. Moreover, the use of an aperiodic sparse array can further reduce the number of sensors efficiently. In this paper, an optimized method for sparse cross array synthesis is proposed. First, the beamforming of a cross array based on a multi-frequency algorithm is simplified for both near-field and far-field. Next, a perturbed convex optimization algorithm is proposed for sparse cross array synthesis. The method based on convex optimization utilizes a first-order Taylor expansion to create position perturbations that can optimize the beam pattern and minimize the number of active sensors. Finally, a cross array with 100 + 100 sensors is employed from which a sparse cross array with 45 + 45 sensors is obtained via the proposed method. The experimental results show that the proposed method is more effective than existing methods for obtaining optimum results for sparse cross array synthesis in both the near-field and far-field.

This paper studies an inexact perturbed path-following algorithm in the framework of Lagrangian dual decomposition for solving large-scale separable convex programming problems. Unlike the exact versions considered in the literature, we propose solving the primal subproblems inexactly up to a given accuracy. This leads to an inexactness of the gradient vector and the Hessian matrix of the smoothed dual function. Then an inexact perturbed algorithm is applied to minimize the smoothed dual function. The algorithm consists of two phases, and both make use of the inexact derivative information of the smoothed dual problem. The convergence of the algorithm is analyzed, and the worst-case complexity is estimated. As a special case, an exact path-following decomposition algorithm is obtained and its worst-case complexity is given. Implementation details are discussed, and preliminary numerical results are reported. [PUBLICATION ABSTRACT]

Recently, Cánovas et al. presented an interesting result: the argmin mapping of a linear semi-infinite program under canonical perturbations is calm if and only if some associated linear semi-infinite inequality system is calm. Using classical tools from parametric optimization, we show that the if-direction of this condition holds in a much more general framework of optimization models, while the opposite direction may fail in the general case. In applications to special classes of problems, we apply a more recent result on the intersection of calm multifunctions.

This study considers the control problem for a class of non-linear singularly perturbed systems (SPSs) subject to actuator saturation. A sufficient condition for the existence of state-feedback controllers to achieve a prescribed stability bound is proposed and the corresponding basin of attraction is estimated. Then a convex optimisation problem is formulated, by which an optimal controller can be obtained to achieve a prescribed stability bound and simultaneously maximise the estimate of the basin of attraction of the SPSs for any allowable singular perturbation parameter. Furthermore, a stability condition is established, which improves the existing stability bound analysis methods for a class of non-linear SPSs. Finally, examples are given to show the advantages and effectiveness of the obtained results.

Biological structure and function depend on complex regulatory interactions between many genes. A wealth of gene expression data is available from high-throughput genome-wide measurement technologies, but effective gene regulatory network inference methods are still needed. Model-based methods founded on quantitative descriptions of gene regulation are among the most promising, but many such methods rely on simple, local models or on ad hoc inference approaches lacking experimental interpretability. We propose an experimental design and develop an associated statistical method for inferring a gene network by learning a standard quantitative, interpretable, predictive, biophysics-based ordinary differential equation model of gene regulation. We fit the model parameters using gene expression measurements from perturbed steady-states of the system, like those following overexpression or knockdown experiments. Although the original model is nonlinear, our design allows us to transform it into a convex optimization problem by restricting attention to steady-states and using the lasso for parameter selection. Here, we describe the model and inference algorithm and apply them to a synthetic six-gene system, demonstrating that the model is detailed and flexible enough to account for activation and repression as well as synergistic and self-regulation, and the algorithm can efficiently and accurately recover the parameters used to generate the data.

In this paper, without using any regularity assumptions, we derive a new exact formula for computing the Fréchet coderivative and an exact formula for the Mordukhovich coderivative of normal cone mappings to perturbed polyhedral convex sets. Our development establishes generalizations and complements of the existing results on the topic. An example to illustrate formulae is given.

The authors investigate the problem of minimizing the perturbed convex function ... over some convex subset D of a normed linear space X, where the function f is convex and the perturbation p is bounded. The key tool for our investigation is a convexity modulus of f named ... , whose generalized inverse function ... is used to define the quantity ... . Generally, by the irregular perturbation p, the perturbed function ... loses all usual analytical and optimization properties yielded by the convexity of f. Moreover, the diameter of the set of global infimizers (including global minimizers) of ... is not greater than ... , and the distance between any global infimizer of ... and any global infimizer of f cannot exceed ... . The latter property is used for sensibility analysis. (ProQuest: ... denotes formulae/symbols omitted.)

The quickly moving market data in the finance industry requires a frequent parameter identification of the corresponding financial market models. In this paper we apply a special sequential quadratic programming algorithm to the calibration of typical equity market models. As it turns out, the projection of the iterates onto the feasible set can be efficiently computed by solving a semidefinite programming problem. Combining this approach with a Gauss-Newton framework leads to an efficient algorithm which allows to calibrate e.g. Heston’s stochastic volatility model in less than a half second on a usual 3 GHz desktop PC. Furthermore we present an appropriate regularization technique that stabilizes and significantly speeds up computations if the model parameters are chosen to be time-dependent.

We analyze the transition and convergence properties of genetic algorithms (GAs) applied to fitness functions perturbed concurrently by additive and multiplicative noise. Both additive noise and multiplicative noise are assumed to take on finitely many values. We explicitly construct a Markov chain that models the evolution of GAs in this noisy environment and analyze it to investigate the algorithms. Our analysis shows that this Markov chain is indecomposable; it has only one positive recurrent communication class. Using this property, we establish a condition that is both necessary and sufficient for GAs to eventually (i.e., as the number of iterations goes to infinity) find a globally optimal solution with probability 1. Similarly, we identify a condition that is both necessary and sufficient for the algorithms to eventually with probability 1 fail to find any globally optimal solution. Our analysis also shows that the chain has a stationary distribution that is also its steady-state distribution. Based on this property and the transition probabilities of the chain, we compute the exact probability that a GA is guaranteed to select a globally optimal solution upon completion of each iteration.

This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions with bounded effective domains defined on a Banach space with the Radon-Nikodým property; and gives an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition.