Nonsmooth **Optimization** over Stiefel Manifold: Riemannian Subgradient
Methods

**optimization**

**problems**over Stefiel manifold, which are ubiquitous in engineering applications but still largely unexplored. We study this type of nonconvex

**optimization**

**problems**under the settings that the function is weakly convex in Euclidean space and locally Lipschitz continuous, where we propose to address these

**problems**using a family of Riemannian subgradient methods. First, we show iteration complexity ${\cal O}(\varepsilon^{-4})$ for these algorithms driving a natural stationary measure to be smaller than $\varepsilon$. Moreover, local linear convergence can be achieved for Riemannian subgradient and incremental subgradient methods if the

**optimization**

**problem**further satisfies the sharpness property and the algorithms are initialized close to the set of weak sharp minima. As a result, we provide the first convergence rate guarantees for a family of Riemannian subgradient methods utilized to optimize nonsmooth functions over Stiefel manifold, under reasonable regularities of the functions. The fundamental ingredient for establishing the aforementioned convergence results is that any weakly convex function in Euclidean space admits an important property holding uniformly over Stiefel manifold which we name Riemannian subgradient inequality. We then extend our convergence results to a broader class of compact Riemannian manifolds embedded in Euclidean space. Finally, we discuss the sharpness property for robust subspace recovery and orthogonal dictionary learning, and demonstrate the established convergence performance of our algorithms on both

**problems**via numerical simulations.

6/10 relevant

arXiv

A Molecular Computing Approach to Solving **Optimization** **Problems** via Programmable Microdroplet Arrays

**optimization**

**problems**are mapped to an Ising Hamiltonian and encoded in the form of intra- and inter- droplet interactions. Expand abstract.

**problems**to be solved. By using droplets and room-temperature processes, molecular computing is a promising research direction with potential biocompatibility and cost advantages. In this work, we present a new approach for computation using a network of chemical reactions taking place within an array of spatially localized droplets whose contents represent bits of information. Combinatorial

**optimization**

**problems**are mapped to an Ising Hamiltonian and encoded in the form of intra- and inter- droplet interactions. The

**problem**is solved by initiating the chemical reactions within the droplets and allowing the system to reach a steady-state; in effect, we are annealing the effective spin system to its ground state. We propose two implementations of the idea, which we ordered in terms of increasing complexity. First, we introduce a hybrid classical-molecular computer where droplet properties are measured and fed into a classical computer. Based on the given

**optimization**problem, the classical computer then directs further reactions via optical or electrochemical inputs. A simulated model of the hybrid classical-molecular computer is used to solve boolean satisfiability and a lattice protein model. Second, we propose architectures for purely molecular computers that rely on pre-programmed nearest-neighbour inter-droplet communication via energy or mass transfer.

10/10 relevant

chemRxiv

Sufficient optimality conditions in bilevel programming

**optimization**

**problems**involving smooth functions. Expand abstract.

**optimization**

**problems**involving smooth functions. First-order sufficient optimality conditions are obtained by estimating the tangent cone to the feasible set of the bilevel program in terms of initial

**problem**data. This is done by exploiting several different reformulations of the hierarchical model as a single-level

**problem**. To obtain second-order sufficient optimality conditions, we exploit the so-called value function reformulation of the bilevel

**optimization**problem, which is then tackled with the aid of second-order directional derivatives. The resulting conditions can be stated in terms of initial

**problem**data in several interesting situations comprising the settings where the lower level is linear or possesses strongly stable solutions.

6/10 relevant

arXiv

A fast two-point gradient algorithm based on sequential subspace
**optimization** method for nonlinear ill-posed **problems**

**problems**, which is based on the sequential subspace

**optimization**method. Expand abstract.

**optimization**method. A complete convergence analysis is provided under the classical assumptions for iterative regularization methods. The design of the two-point gradient method involves the choices of the combination parameters which is systematically discussed. Furthermore, detailed numerical simulations are presented for inverse potential problem, which exhibit that the proposed method leads to a strongly decrease of the iteration numbers and the overall computational time can be significantly reduced.

4/10 relevant

arXiv

Robust Control **Optimization** for Quantum Approximate Optimization
Algorithm

**optimization**algorithm (QAOA), which can be used to solve certain quantum control

**problems**, state preparation

**problem**s, and combinatorial optimization problems. Expand abstract.

**optimization**algorithm (QAOA), which can be used to solve certain quantum control problems, state preparation problems, and combinatorial

**optimization**

**problems**. We demonstrate that the error of QAOA simulation can be significantly reduced by robust control

**optimization**techniques, specifically, by sequential convex programming (SCP), to ensure error suppression in situations where the source of the error is known but not necessarily its magnitude. We show that robust

**optimization**improves both the objective landscape of QAOA as well as overall circuit fidelity in the presence of coherent errors and errors in initial state preparation.

4/10 relevant

arXiv

Training Neural Networks for Likelihood/Density Ratio Estimation

**optimization**

**problems**with guarantees that the solution is indeed the desired function. Expand abstract.

**problems**in Engineering and Statistics require the computation of the likelihood ratio function of two probability densities. In classical approaches the two densities are assumed known or to belong to some known parametric family. In a data-driven version we replace this requirement with the availability of data sampled from the densities of interest. For most well known

**problems**in Detection and Hypothesis testing we develop solutions by providing neural network based estimates of the likelihood ratio or its transformations. This task necessitates the definition of proper

**optimizations**which can be used for the training of the network. The main purpose of this work is to offer a simple and unified methodology for defining such

**optimization**

**problems**with guarantees that the solution is indeed the desired function. Our results are extended to cover estimates for likelihood ratios of conditional densities and estimates for statistics encountered in local approaches.

5/10 relevant

arXiv

Min-Max-Min Robustness for Combinatorial **Problems** with Discrete Budgeted
Uncertainty

**problem**with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min-max-min problem is NPhard for many easy combinatorial

**optimization**

**problems**, and not approximable in general. Expand abstract.

**optimization**

**problems**with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min-max-min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage

**problems**. The uncertainty set assumed in this paper considers that in any scenario, at most Gamma of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set. While the classical min-max

**problem**with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min-max-min

**problem**is NPhard for many easy combinatorial

**optimization**problems, and not approximable in general. We thus present an integer programming formulation for solving the

**problem**through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times.

5/10 relevant

arXiv

A Framework for Data-Driven Computational Mechanics Based on Nonlinear
**Optimization**

**optimization**

**problem**, which can be robustly solved, is computationally efficient, and does not rely on any special functional structure of the reconstructed constitutive manifold. Expand abstract.

**optimization**

**problem**and seeks to assign to each material point a point in the phase space that satisfies compatibility and equilibrium, while being closest to the data set provided. The second one is a data driven inverse approach that seeks to reconstruct a constitutive manifold from data sets by manifold learning techniques, relying on a well-defined functional structure of the underlying constitutive law. In this work, we propose a third route that combines the strengths of the two existing directions and mitigates some of their weaknesses. This is achieved by the formulation of an approximate nonlinear

**optimization**problem, which can be robustly solved, is computationally efficient, and does not rely on any special functional structure of the reconstructed constitutive manifold. Additional benefits include the natural incorporation of kinematic constraints and the possibility to operate with implicitly defined stress-strain relations. We discuss important mathematical aspects of our approach for a data-driven truss element and investigate its key numerical behavior for a data-driven beam element that makes use of all components of our methodology.

4/10 relevant

arXiv

Solving **Optimization** **Problems** through Fully Convolutional Networks: an
Application to the Travelling Salesman **Problem**

**optimization**

**problem**through deep learning? Expand abstract.

**optimization**algorithms greatly contribute to the development of deep learning. But the reverse applications are still insufficient. Is there any efficient way to solve certain

**optimization**

**problem**through deep learning? The key is to convert the

**optimization**to a representation suitable for deep learning. In this paper, a traveling salesman

**problem**(TSP) is studied. Considering that deep learning is good at image processing, an image representation method is proposed to transfer a TSP to an image. Based on samples of a 10 city TSP, a fully convolutional network (FCN) is used to learn the mapping from a feasible region to an optimal solution. The training process is analyzed and interpreted through stages. A visualization method is presented to show how a FCN can understand the training task of a TSP. Once the training is completed, no significant effort is required to solve a new TSP and the prediction is obtained on the scale of milliseconds. The results show good performance in finding the global optimal solution. Moreover, the developed FCN model has been demonstrated on TSP's with different city numbers, proving excellent generalization performance.

9/10 relevant

arXiv

Convergent Policy **Optimization** for Safe Reinforcement Learning

**problem**, we construct a sequence of surrogate convex constrained

**optimization**

**problems**by replacing the nonconvex functions locally with convex quadratic functions obtained from policy gradient estimators. Expand abstract.

**problem**with nonlinear function approximation, where policy

**optimization**is formulated as a constrained

**optimization**

**problem**with both the objective and the constraint being nonconvex functions. For such a problem, we construct a sequence of surrogate convex constrained

**optimization**

**problems**by replacing the nonconvex functions locally with convex quadratic functions obtained from policy gradient estimators. We prove that the solutions to these surrogate

**problems**converge to a stationary point of the original nonconvex

**problem**. Furthermore, to extend our theoretical results, we apply our algorithm to examples of optimal control and multi-agent reinforcement learning with safety constraints.

7/10 relevant

arXiv