Set Of Local Minimizers Research Articles

On Landscape of Nonconvex Regularized Least Squares for Sparse Support Recovery

Exact sparse support recovery from noisy deterministic Fourier measurements is studied in this paper. The ideal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> -regularized least squares (LS) program is known to have strict local minimizers for multiple sparse supports. By applying a specific non-convex surrogate of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula> -regularizer, we show that, under certain conditions, the set of local minimizers with particularly separated supports is a singleton. Thus, any converging algorithms that can enforce the corresponding separation constraint will lead to the true support as the unique solution. In this regard, we construct a novel optimal mapping operator that can fulfill the separation condition, and several empirically effective algorithms are developed. The simulation results demonstrate the theoretical claims made in this paper.

On the local and global minimizers of gradient regularized model with box constraints for image restoration

Recently, nonconvex and nonsmooth models such as those using ‘norm’ have drawn much attention in the area of image restoration. This work investigates the local and global minimizers of the gradient regularized model with box constraints. There are four major ingredients. Firstly, we show that the set of local minimizers can be represented by solutions to some quadratic problems, which are independent of the fidelity parameter α. Based on this, every point satisfying the first-order necessary condition is a local minimizer. Secondly, any two local minimizers have different energy values under certain assumptions, implying the uniqueness of the global minimizer. Thirdly, there exists a uniform lower bound for nonzero gradients of the restored images. Finally, we show that the global minimizer set is piecewise constant in terms of α, and when A is of full column rank and α is large enough, the distance between the true image and the restored images is bounded by the noise level. The numerical examples perfectly demonstrate our theoretical analysis.

Multiobjective optimization of ceramic-metal functionally graded plates using a higher order model

A methodology of multiobjective design optimization of ceramic–metal composite plates with functionally graded materials, with properties varying through the thickness direction, obtained by an adequate variation of volume fractions of the constituent materials, is presented in this paper. Constrained optimization is conducted for different behavior objectives like the maximization of buckling load or fundamental natural frequency. Mass minimization and material cost minimization are also considered. The optimization problems are constrained by stress based failure criteria and other structural response constraints or manufacturing limitations. The design variables are the index of the power-law distribution in the metal-ceramic graded material and the thicknesses of the graded material and/or the metal and ceramic faces.An equivalent single layer finite element plate model having a displacement field based on a higher order shear deformation theory, accounting for the temperature dependency of the material properties, was developed and validated for the analysis of through-the-thickness ceramic-metal functionally graded plates. The optimization problems are solved with two direct search derivative-free algorithms: GLODS (Global and Local Optimization using Direct Search) and DMS (Direct MultiSearch). DMS, the multiobjective optimization solver, is started from a set of local minimizers which are initially determined by the global optimizer algorithm GLODS for each one of the objective functions.

Multiobjective design optimization of laminated composite plates with piezoelectric layers

A methodology of multiobjective design optimization of laminated composite plates with piezoelectric layers is presented in this paper. Constrained optimization is conducted for different behaviour objectives, like the maximization of buckling load or natural frequencies of specific vibration modes or prescribed displacements for example. Weight minimization can also be considered or the minimization of the electric voltages applied in the piezoelectric actuators. The optimization problems are constrained by stress based failure criteria and other structural response constraints like limits imposed on certain displacements, buckling characteristics and natural frequency constraints. The design variables considered in the present work are the fiber reinforcement orientations in the composite layers, thicknesses of individual layers and the electric potentials applied to the actuators.The optimization problems are solved with two direct search derivative-free algorithms: GLODS (Global and Local Optimization using Direct Search) and DMS (Direct MultiSearch). DMS, the multiobjective optimization solver, is started from a set of local minimizers which are initially determined by the global optimizer algorithm GLODS for each one of the objective functions.

Sparsifying Transform Learning With Efficient Optimal Updates and Convergence Guarantees

Many applications in signal processing benefit from the sparsity of signals in a certain transform domain or dictionary. Synthesis sparsifying dictionaries that are directly adapted to data have been popular in applications such as image denoising, inpainting, and medical image reconstruction. In this work, we focus instead on the sparsifying transform model, and study the learning of well-conditioned square sparsifying transforms. The proposed algorithms alternate between a $\ell_0$ "norm"-based sparse coding step, and a non-convex transform update step. We derive the exact analytical solution for each of these steps. The proposed solution for the transform update step achieves the global minimum in that step, and also provides speedups over iterative solutions involving conjugate gradients. We establish that our alternating algorithms are globally convergent to the set of local minimizers of the non-convex transform learning problems. In practice, the algorithms are insensitive to initialization. We present results illustrating the promising performance and significant speed-ups of transform learning over synthesis K-SVD in image denoising.

Newton's problem of the body of minimal resistance under a single-impact assumption

We consider the problem of the body of minimal resistance as formulated in [2], Sect. 5: minimize \(F(u):=\int_\Omega dx/(1+|\nabla u(x)|^2)\), where \(\Omega\) is the unit disc of \({\mathbb R}^2\), in the class of radial functions \(u:\Omega\to[0,M]\) satisfying a geometrical property (1), corresponding to a single-impact assumption (\(M>0\) is a given parameter). We prove the existence of a critical value \(M^*\) of M. For \(M\geq M^*\), there exist a unique local minimizer of the functional. For \(M< M^*\), the set of local minimizers is not compact in \(H^1\), though they all achieve the same value of the functional.

On the best case performance of hit and run methods for detecting necessary constraints

The hit and run methods are probabilistic algorithms that can be used to detect necessary (nonredundant) constraints in systems of linear constraints. These methods construct random sequences of lines that pass through the feasible region. These lines intersect the boundary of the region at twohit-points, each identifying a necessary constraint. In order to study the statistical performance of such methods it is assumed that the probabilities of hitting particular constraints are the same for every iteration. An indication of the best case performance of these methods can be determined by minimizing, with respect to the hit probabilities, the expected value of the number of iterations required to detect all necessary constraints. We give a set of isolated strong local minimizers and prove that for two, three and four necessary constraints the set of local minimizers is the complete set of global minimizers. We conjecture that this is also the case for any number of necessary constraints. The results in this paper also apply to sampling problems (e.g., balls from an urn) and to the coupon collector's problem.

Lipschitzian Solutions of Perturbed Nonlinear Programming Problems

We prove that if a second order sufficient condition and a constraint regularity assumption hold, then for sufficiently small perturbations of the constraints and the objective function, the set of local minimizers reduces to a singleton. Moreover, the minimizes and the associated multipliers are Lipschitzian functions of the parameter.