Non-strictly Convex Research Articles

A fast trajectory planning framework with task-priority for space robot

This paper addresses trajectory planning issue for free-floating space robotic system which is designed to track a prescribed path simultaneously fulfilling imposed constraints such as collision avoidance and minimizing base attitude disturbance. Collision avoidance can be treated as inequality constraints as part of the trajectory planning. To generate smooth trajectory for robot motion, a novel local collision avoidance method considering multiple velocity damper constraints is proposed for non-strictly convex polyhedra. Moreover, we propose a task-priority handling strategy incorporated with a full-fledged Quadratic Programming(QP) procedure to solve the planning problem with sequential tasks. The proposed task-priority handling strategy enables a rapid adaptation of the space manipulators motion, which makes this strategy feasible for real-time planning. The trajectory planning framework with collision avoidance method and task-priority handling strategy were implemented and demonstrated via a kinematically redundant dual-arm space robotic system.

A Numerically Stable Solver for Positive Semidefinite Quadratic Programs Based on Nonnegative Least Squares

This paper proposes a new algorithm for solving convex quadratic programs (QP) subject to linear inequality and equality constraints. The method extends an approach recently proposed by the author for solving strictly convex QP's using nonnegative least squares, by making it numerically more robust and able to handle also the nonstrictly convex case, equality constraints, and warm starting from an initial guess. Robustness is achieved by introducing an outer proximal-point iteration scheme that regularizes the problem without altering the solution, and by adaptively weighting the least squares problems encountered while solving the problem. The performance of the resulting QP solver in terms of speed and robustness in the single precision arithmetic is assessed against other optimization algorithms tailored to embedded model predictive control applications.

A minimization approach to conservation laws with random initial conditions and non-smooth, non-strictly convex flux

We obtain solutions to conservation laws under any random initial conditions that are described by Gaussian stochastic processes (in some cases discretized). We analyze the generalization of Burgers' equation for a smooth flux function $H\left( p\right) =\left\vert p\right\vert ^{j}$ for $j\geq2$ under random initial data. We then consider a piecewise linear, non-smooth and non-convex flux function paired with general discretized Gaussian stochastic process initial data. By partitioning the real line into a finite number of points, we obtain an exact expression for the solution of this problem. From this we can also find exact and approximate formulae for the density of shocks in the solution profile at a given time $t$ and spatial coordinate $x$. We discuss the simplification of these results in specific cases, including Brownian motion and Brownian bridge, for which the inverse covariance matrix and corresponding eigenvalue spectrum have some special properties. We calculate the transition probabilities between various cases and examine the variance of the solution $w\left(x,t\right)$ in both $x$ and $t$. We also describe how results may be obtained for a non-discretized version of a Gaussian stochastic process by taking the continuum limit as the partition becomes more fine.

Minimization solutions to conservation laws with non-smooth and non-strictly convex flux

Conservation laws are usually studied in the context of sufficient regularity conditions imposed on the flux function, usually $C^{2}$ and uniform convexity. Some results are proven with the aid of variational methods and a unique minimizer such as Hopf-Lax and Lax-Oleinik. We show that many of these classical results can be extended to a flux function that is not necessarily smooth or uniformly or strictly convex. Although uniqueness a.e. of the minimizer will generally no longer hold, by considering the greatest (or supremum, where applicable) of all possible minimizers, we can successfully extend the results. One specific nonlinear case is that of a piecewise linear flux function, for which we prove existence and uniqueness results. We also approximate it by a smoothed, superlinearized version parameterized by $\varepsilon$ and consider the characterization of the minimizers for the smooth version and limiting behavior as $\varepsilon\downarrow0$ to that of the sharp, polygonal problem. In proving a key result for the solution in terms of the value of the initial condition, we provide a stepping stone to analyzing the system under stochastic processes, which will be explored further in a future paper.

Limit theory for random walks in degenerate time-dependent random environments

We study continuous-time (variable speed) random walks in random environments on Zd, d≥2, where, at time t, the walk at x jumps across edge (x,y) at time-dependent rate at(x,y). The rates, which we assume stationary and ergodic with respect to space–time shifts, are symmetric and bounded but possibly degenerate in the sense that the total jump rate from a vertex may vanish over finite intervals of time. We formulate conditions on the environment under which the law of diffusively-scaled random walk path tends to Brownian motion for almost every sample of the rates. The proofs invoke Moser iteration to prove sublinearity of the corrector in pointwise sense; a key additional input is a conversion of certain weighted energy norms to ordinary ones. Our conclusions apply to random walks on dynamical bond percolation and interacting particle systems as well as to random walks arising from the Helffer–Sjöstrand representation of gradient models with certain non-strictly convex potentials.

An Alternating Direction Method Approach to Cloud Traffic Management

In this paper, we introduce a unified framework for studying various cloud traffic management problems, ranging from geographical load balancing to backbone traffic engineering. We first abstract these real-world problems as a multi-facility resource allocation problem, and then present two distributed optimization algorithms by exploiting the special structure of the problem. Our algorithms are inspired by Alternating Direction Method of Multipliers (ADMM), enjoying a number of unique features. Compared to dual decomposition, they converge with non-strictly convex objective functions; compared to other ADMM-type algorithms, they not only achieve faster convergence under weaker assumptions, but also have lower computational complexity and lower message-passing overhead. The simulation results not only confirm these desirable features of our algorithms, but also highlight several additional advantages, such as scalability and fault-tolerance.

Factorisable Multi-Task Quantile Regression

For many applications, analyzing multiple response variables jointly is desirable because of their dependency, and valuable information about the distribution can be retrieved by estimating quantiles. In this paper, we propose a multi-task quantile regression method that exploits the potential factor structure of multivariate conditional quantiles through nuclear norm regularization. We jointly study the theoretical properties and computational aspects of the estimating procedure. In particular, we develop an efficient iterative proximal gradient algorithm for the non-smooth and non-strictly convex optimization problem incurred in our estimating procedure, and derive oracle bounds for the estimation error in a realistic situation where the sample size and number of iterative steps are both finite. The finite iteration analysis is particular useful when the matrix to be estimated is big and the computational cost is high. Merits of the proposed methodology are demonstrated through a Monte Carlo experiment and applications to climatological and financial study. Specifically, our method provides an objective foundation for spatial extreme clustering, and gives a refreshing look on the global financial systemic risk. Supplementary materials for this article are available online.

Contractibility of Fixed Point Sets of Mean-Type Mappings

We establish a convergence theorem and explore fixed point sets of certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces. We not only extend previous works by Matkowski to general normed linear spaces, but also obtain a new result on the structure of fixed point sets of quasi-nonexpansive mappings in a nonstrictly convex setting.

Sub-Finsler Structures from the Time-Optimal Control Viewpoint for some Nilpotent Distributions

In this paper, we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.

A uniqueness result for a class of non-strictly convex variational problems

Let Ω be a smooth domain in R2, we prove that if g:[0,+∞)→[0,+∞] is convex with g(0)<g(t) whenever t>0 then there exists an unique minimizer u∈C0,1(Ω) of the functional u↦∫Ωg(|∇u|)dxdy among all Lipschitz-continuous functions that assume the same value of u on ∂Ω.

Steinhaus' lattice-point problem for Banach spaces

Steinhaus proved that given a positive integer n, one may find a circle surrounding exactly n points of the integer lattice. This statement has been recently extended to Hilbert spaces by Zwoleński, who replaced the integer lattice by any infinite set that intersects every ball in at most finitely many points. We investigate Banach spaces satisfying this property, which we call (S), and characterise them by means of a new geometric property of the unit sphere which allows us to show, e.g., that all strictly convex norms have (S), nonetheless, there are plenty of non-strictly convex norms satisfying (S). We also study the corresponding renorming problem.

Computation Algorithm for Convex Semi-infinite Program with Second-Order Cones: Special Analyses for Affine and Quadratic Case

We focus on the convex semi-infinite program with second-order cone constraints (for short, SOCCSIP), which has wide applications such as filter design, robust optimization, and so on. For solving the SOCCSIP, we propose an explicit exchange method, and prove that the algorithm terminates in a finite number of iterations. In the convergence analysis, we do not need to use the special structure of second-order cone (SOC) when the objective or constraint function is strictly convex. However, if both of them are non-strictly convex and constraint function is affine or quadratic, then we have to utilize the SOC complementarity conditions and the spectral factorization techniques associated with Euclidean Jordan algebra. We also show that the obtained output is an approximate optimum of SOCCSIP. We report some numerical results involving the application to the robust optimization in the classical convex semi-infinite program.

Solving the Pareto front for multiobjective Markov chains using the minimum Euclidean distance gradient-based optimization method

A novel method based on minimizing the Euclidean distance is proposed for generating a well-distributed Pareto set in multi-objective optimization for a class of ergodic controllable Markov chains. The proposed approach is based on the concept of strong Pareto policy. We consider the case where the search space is a non-strictly convex set. For solving the problem we introduce the Tikhonov’s regularization method and implement the Lagrange principle. We formulate the original problem introducing linear constraints over the nonlinear problem employing the c-variable method and constraining the cost-functions allowing points in the Pareto front to have a small distance from one another. As a result, the proposed method generates an even representation of the entire Pareto surface. Then, we propose an algorithm to compute the Pareto front and provide all the details needed to implement the method in an efficient and numerically stable way. As well, we prove the main Theorems for describing the dependence of the saddle point for the regularizing parameter and analyzes its asymptotic behavior. Moreover, we analyze the step size parameter of the Lagrange principle and also its asymptotic behavior. The suggested approach is validated theoretically and verified by a numerical example related to security patrolling that present a technique for visualizing the Pareto front.

Hierarchical Convex Optimization With Primal-Dual Splitting

This paper addresses the selection of a desirable solution among all the solutions of a convex optimization problem (referred to as the first-stage problem) mainly for inverse problems in signal processing. This is realized in the framework of hierarchical convex optimization, i.e., minimizing another convex function over the solution set of the first-stage problem. Hierarchical convex optimization is an ideal strategy when the first-stage problem has infinitely many solutions because of the non-strict convexity of its objective function, which could arise in various scenarios, e.g., convex feasibility problems. To this end, first, the fixed point set characterization behind a primal-dual splitting type method is incorporated into the framework of hierarchical convex optimization, which enables the framework to cover a broad class of first-stage problem formulations. Then, a pair of efficient algorithmic solutions to the hierarchical convex optimization problem, as certain realizations of the hybrid steepest descent method, are provided with guaranteed convergence. We also present a specialized form of the proposed framework to focus on a typical scenario of inverse problems, and show its application to signal interpolation.

Symmetric Constrained Optimal Control

This paper extends previous results on symmetry in strictly convex linear model predictive control to non-strictly convex and nonlinear model predictive control. We define symmetry for constrained systems, controllers, and model predictive control problems. We show that symmetric model predictive control problems produce symmetric controllers. We show that the previously established methods of memory reduction can be applied to non-strictly convex problems. We apply these memory reduction techniques to the battery balancing problem. Exploiting symmetry leads to an exponential memory reduction and simple, intuitive optimal controllers.

A nonstandard free boundary problem arising in the shape optimization of thin torsion rods

We study a 2 d -variational problem, in which the cost functional is an integral depending on the gradient through a convex but not strictly convex integrand, and the admissible functions have zero gradient on the complement of a given domain D . We are interested in establishing whether solutions exist whose gradient “avoids” the region of non-strict convexity. Actually, the answer to this question is related to establishing whether homogenization phenomena occur in optimal thin torsion rods. We provide some existence results for different geometries of D , and we study the nonstandard free boundary problem with a gradient obstacle, which is obtained through the optimality conditions.

Two properties of two-velocity two-pressure models for two-phase flows

We study a class of models of compressible two-phase flows. This class, which includes the Baer-Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly convex and that the system admits a symmetric form.

Large deviations for renewal processes

We investigate large deviations for the empirical measure of the forward and backward recurrence time processes associated with a classical renewal process with arbitrary waiting-time distribution. The Donsker–Varadhan theory cannot be applied in this case, and indeed it turns out that the large deviations rate functional differs from the one suggested by such a theory. In particular, a non-strictly convex and non-analytic rate functional is obtained.

METHOD OF DETERMINATION OF THE FIRST PLAYER OPTIMAL STRATEGIES IN A SUBCLASS OF THE NONSTRICTLY CONVEX ANTAGONISTIC GAMES

By the example of two nonstrictly convex antagonistic games, where the second player has the single optimal pure strategy, it has been asserted, that there exists a subclass of nonstrictly convex antagonistic games, in which by the known method there cannot be determined the optimal probabilities of selecting the essential pure strategies of the first player. It has been demonstrated that to determine them, it is sufficient to employ the saddle point concept in the known optimality principle by applying the corresponding right-side inequality.

A strategy for non-strictly convex transport costs and the example of $║x-y║^p$ in $R^2$

This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non-strictly convex cost. We give a decomposition strategy to address this issue. As a consequence of our procedure, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. To illustrate possible results obtained through this general approach, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form $h║x-y║$, with h strictly convex increasing and $║.║$ an arbitrary norm in $R^2$.