Alfonso: Matlab Package for Nonsymmetric Conic Optimization

We present alfonso, an open-source Matlab package for solving conic optimization problems over nonsymmetric convex cones. The implementation is based on the authors’ corrected analysis of a method of Skajaa and Ye. It enables optimization over any convex cone as long as a logarithmically homogeneous self-concordant barrier is available for the cone or its dual. This includes many nonsymmetric cones, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, algorithms for nonsymmetric conic optimization also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. The worst-case iteration complexity of alfonso is the best known for nonsymmetric cone optimization: [Formula: see text] iterations to reach an ε-optimal solution, where ν is the barrier parameter of the barrier function used in the optimization. Alfonso can be interfaced with a Matlab function (supplied by the user) that computes the Hessian of a barrier function for the cone. A simplified interface is also available to optimize over the direct product of cones for which a barrier function has already been built into the software. This interface can be easily extended to include new cones. Both interfaces are illustrated by solving linear programs. The oracle interface and the efficiency of alfonso are also demonstrated using an optimal design of experiments problem in which the tailored barrier computation greatly decreases the solution time compared with using state-of-the-art, off-the-shelf conic optimization software. Summary of Contribution: The paper describes an open-source Matlab package for optimization over nonsymmetric cones. A particularly important feature of this software is that, unlike other conic optimization software, it enables optimization over any convex cone as long as a suitable barrier function is available for the cone or its dual, not limiting the user to a small number of specific cones. Nonsymmetric cones for which such barriers are already known include, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Thus, the scope of this software is far larger than most current conic optimization software. This does not come at the price of efficiency, as the worst-case iteration complexity of our algorithm matches the iteration complexity of the most successful interior-point methods for symmetric cones. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, our software can also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. This is also demonstrated in this paper via an example in which our code significantly outperforms Mosek 9 and SCS 2.

Siegel domains over Finsler symmetric cones

Let Ω be a proper open cone in a real Banach space V. We show that the tube domain V ⊕ i ⁢ Ω {V\oplus i\Omega} over Ω is biholomorphic to a bounded symmetric domain if and only if Ω is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that V is a unital JB-algebra in an equivalent norm and Ω is the interior of { v 2 : v ∈ V } {\{v^{2}:v\in V\}} .

The Elastic Flow with Obstacles: Small Obstacle Results

We consider a parabolic obstacle problem for Euler’s elastic energy of graphs with fixed ends. We show global existence, well-posedness and subconvergence provided that the obstacle and the initial datum are suitably ‘small’. For symmetric cone obstacles we can improve the subconvergence to convergence. Qualitative aspects such as energy dissipation, coincidence with the obstacle and time regularity are also examined.

Uniaxial Compression Mechanical Analysis of Cylindrical Rock Specimens Based on Space Axisymmetric Problem

The common failure forms in the uniaxial compression test of standard cylindrical rock specimens are symmetric cone failure and splitting failure. In the past, two kinds of failure forms were explained from the plane problems of elasticity theory. However, there was a lack of research based on space problems. In this paper, based on spatial axisymmetric elastic mechanics theory, three-dimensional stress function is constructed for the cylindrical rock specimen under uniform axial compression by adopting the semi-inverse method. According to the stress function, the stress components and the principal stress are deduced when considering the end effect. The failure form is speculated, which can well explain the two common types of damage in uniaxial compression. This work may enrich the basic research of rock mechanics.

A primal-dual interior-point algorithm for nonsymmetric exponential-cone optimization

A new primal-dual interior-point algorithm applicable to nonsymmetric conic optimization is proposed. It is a generalization of the famous algorithm suggested by Nesterov and Todd for the symmetric conic case, and uses primal-dual scalings for nonsymmetric cones proposed by Tunçel. We specialize Tunçel’s primal-dual scalings for the important case of 3 dimensional exponential-cones, resulting in a practical algorithm with good numerical performance, on level with standard symmetric cone (e.g., quadratic cone) algorithms. A significant contribution of the paper is a novel higher-order search direction, similar in spirit to a Mehrotra corrector for symmetric cone algorithms. To a large extent, the efficiency of our proposed algorithm can be attributed to this new corrector.

Elastic-instability–enabled locomotion

Locomotion of an organism interacting with an environment is the consequence of a symmetry-breaking action in space-time. Here we show a minimal instantiation of this principle using a thin circular sheet, actuated symmetrically by a pneumatic source, using pressure to change shape nonlinearly via a spontaneous buckling instability. This leads to a polarized, bilaterally symmetric cone that can walk on land and swim in water. In either mode of locomotion, the emergence of shape asymmetry in the sheet leads to an asymmetric interaction with the environment that generates movement––via anisotropic friction on land, and via directed inertial forces in water. Scaling laws for the speed of the sheet of the actuator as a function of its size, shape, and the frequency of actuation are consistent with our observations. The presence of easily controllable reversible modes of buckling deformation further allows for a change in the direction of locomotion in open arenas and the ability to squeeze through confined environments––both of which we demonstrate using simple experiments. Our simple approach of harnessing elastic instabilities in soft structures to drive locomotion enables the design of novel shape-changing robots and other bioinspired machines at multiple scales.