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It is well known that the second-order cone and the circular cone have many analogous properties. In particular, there exists an important distance inequality associated with the second-order cone and the circular cone. The inequality indicates that the distances of arbitrary points to the second-order cone and the circular cone are equivalent, which is crucial in analyzing the tangent cone and normal cone for the circular cone. In this paper, we provide an alternative approach to achieve the aforementioned inequality. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds. Such a clarification is helpful for further study of the relationship between the second-order cone programming problems and the circular cone programming problems.

. Motivated by block partitioned problems arising from group sparsity representation and generalized noncooperative potential games, this paper presents a basic decomposition method for a broad class of multiblock nonsmooth optimization problems subject to coupled linear constraints on the variables that may additionally be individually constrained. The objective of such an optimization problem is given by the sum of two nonseparable functions minus a sum of separable, pointwise maxima of finitely many convex differentiable functions. One of the former two nonseparable functions is of the class LC1 , i.e., differentiable with a Lipschitz gradient, while the other summand is multiconvex. The subtraction of the separable, pointwise maxima of convex functions induces a partial difference-of-convex (DC) structure in the overall objective; yet with all three terms together, the objective is nonsmooth and non-DC, but is blockwise directionally differentiable. By taking advantage of the (negative) pointwise maximum structure in the objective, the developed algorithm and its convergence result are aimed at the computation of a blockwise directional stationary solution, which arguably is the sharpest kind of stationary solutions for this class of nonsmooth problems. This aim is accomplished by combining the alternating direction method of multipliers (ADMM) with a semilinearized Gauss–Seidel scheme, resulting in a decomposition of the overall problem into subproblems each involving the individual blocks. To arrive at a stationary solution of the desired kind, our algorithm solves multiple convex subprograms at each iteration, one per convex function in each pointwise maximum. In order to lessen the potential computational burden in each iteration, a probabilistic version of the algorithm is presented and its almost sure convergence is established.

This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and non-negative constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.

In this paper, we formulate a problem of simultaneous redundant via insertion and line end extension for via yield optimization. Our problem is more general than previous works in the sense that more than one type of line end extension is considered and the objective function to be optimized directly accounts for via yield. We present a zero-one integer linear program based approach, that is equipped with two speedup techniques, to solve the addressed problem optimally. In addition, we describe how to modify our approach to exactly solve a previous work. Extensive experimental results are shown to demonstrate the effectiveness and efficiency of our approaches.

We propose a class of interior proximal-like algorithms for the second-order cone program, which is to minimize a closed proper convex function subject to general second-order cone constraints. The class of methods uses a distance measure generated by a twice continuously differentiable strictly convex function on (0,+\infty), and includes as a special case the entropy-like proximal algorithm [Eggermont, <i>Linear Algebra Appl.</i>, 130 (1990), pp. 2542], which was originally proposed for minimizing a convex function subject to nonnegative constraints. Particularly, we consider an approximate version of these methods, allowing the inexact solution of subproblems. Like the entropy-like proximal algorithm for convex programming with nonnegative constraints, we, under some mild assumptions, establish the global convergence expressed in terms of the objective values for the proposed algorithm, and we show that the

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The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered, and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.

Merit function approach is a popular method to deal with complementarity problems, in which the complementarity problem is recast as an unconstrained minimization via merit function or complementarity function. In this paper, for the complementarity problem associated with <i>p</i>-order cone, which is a type of nonsymmetric cone complementarity problem, we show the readers how to construct merit functions for solving <i>p</i>-order cone complementarity problem. In addition, we study the conditions under which the level sets of the corresponding merit functions are bounded, and we also assert that these merit functions provide an error bound for the <i>p</i>-order cone complementarity problem. These results build up a theoretical basis for the merit method for solving <i>p</i>-order cone complementarity problem.

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