Bilinear Problem Research Articles

Optimal bilinear control problem related to a chemo-repulsion system in 2D domains

In this paper, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a bidimensional domain. The existence, uniqueness and regularity of strong solutions of this model are deduced, proving the existence of a global optimal solution. Afterwards, we derive first-order optimality conditions by using a Lagrange multipliers theorem.

A Convex Solution of the $H_\infty$-Optimal Controller Synthesis Problem for Multidelay Systems

Optimal controller synthesis is a bilinear problem and hence difficult to solve in a computationally efficient manner. We are able to resolve this bilinearity for systems with delay by first convexifying the problem in infinite dimensions, i.e., formulating the $H_\infty$-optimal state-feedback controller synthesis problem for distributed-parameter systems as a linear operator inequality, which a form of convex optimization with operator variables. Next, we use positive matrices to parameterize positive “complete quadratic” operators, allowing the controller synthesis problem to be solved using semidefinite programming (SDP). We then use the solution to this SDP to calculate the feedback gains and provide effective methods for real-time implementation. Finally, we use several test cases to verify that the resulting controllers are optimal to several decimal places as measured by the minimal achievable closed-loop $H_\infty$ norm, and as compared against controllers designed using high-order Padé approximations.

Approximate Nash Equilibrium Solutions of Linear Quadratic Differential Games

It is well known that finding Nash equilibrium solutions of nonzero-sum differential games is a challenging task. Focusing on a class of linear quadratic differential games, we consider three notions of approximate feedback Nash equilibrium solutions and provide a characterisation of these in terms of matrix inequalities which constitute quadratic feasibility problems. These feasibility problems are then recast first as bilinear feasibility problems and finally as rank constrained optimisation problems, i.e. a class of static problems frequently encountered in control theory.

Accelerated First-Order Continuous-Time Algorithm for Solving Convex-Concave Bilinear Saddle Point Problem

First-order methods have simple structures and are of great importance to big data problems because first-order methods are easy to implement in a distributed or parallel way. However, in the worst cases, first-order methods often converge at a rate O(1/t), which is slow. This paper considers a class of convex-concave bilinear saddle point problems and proposes an accelerated first-order continuous-time algorithm. We design the accelerated algorithm by using both increasing and decreasing damping coefficients in the saddle point dynamics. If parameters of the proposed algorithm are proper, the algorithm owns O(1/t2) convergence without any strict or strong convexity requirement. Finally, we apply the algorithm to numerical examples to show the superior performance of the proposed algorithm over existing ones.

A Regularity Criterion for a 3D Chemo-Repulsion System and Its Application to a Bilinear Optimal Control Problem

In this paper, a bilinear optimal control problem associated to a 3D chemo-repulsion model with linear production is studied. The existence of weak solutions is proved, and a regularity criterion t...

Optimal Feedback Controllers for a Stochastic Differential Equation with Reflection

The optimal control problem governed by the stochastic reflection problem associated with a closed convex set $K$ in $\mathbb{R}^d$ is reduced via the corresponding Kolmogorov equation to a deterministic bilinear parabolic optimal control problem on $(0,T)\times K$. In this way, one gets directly a stochastic optimal feedback controller by avoiding the standard dynamic programming equation associated with the stochastic optimal control problem.

Recovering Structured Data From Superimposed Non-Linear Measurements

This work deals with the problem of distributed data acquisition under non-linear communication constraints. More specifically, we consider a model setup where $M$ distributed nodes take individual measurements of an unknown structured source vector $x_0 \in \mathbb{R}^n$, communicating their readings simultaneously to a central receiver. Since this procedure involves collisions and is usually imperfect, the receiver measures a superposition of non-linearly distorted signals. In a first step, we will show that an $s$-sparse vector $x_0$ can be successfully recovered from $O(s \cdot\log(2n/s))$ of such superimposed measurements, using a traditional Lasso estimator that does not rely on any knowledge about the non-linear corruptions. This direct method however fails to work for several "uncalibrated" system configurations. These blind reconstruction tasks can be easily handled with the $\ell^{1,2}$-Group-Lasso, but coming along with an increased sampling rate of $O(s\cdot \max\{M, \log(2n/s) \})$ observations - in fact, the purpose of this lifting strategy is to extend a certain class of bilinear inverse problems to non-linear acquisition. Our two algorithmic approaches are a special instance of a more abstract framework which includes sub-Gaussian measurement designs as well as general (convex) structural constraints. These results are of independent interest for various recovery and learning tasks, as they apply to arbitrary non-linear observation models. Finally, to illustrate the practical scope of our theoretical findings, an application to wireless sensor networks is discussed, which actually serves as the prototypical example of our methodology.

Bilinear Compressed Sensing Under Known Signs via Convex Programming

We consider the bilinear inverse problem of recovering two vectors, $\boldsymbol{x} \in\mathbb{R}^L$ and $\boldsymbol{w} \in\mathbb{R}^L$, from their entrywise product. We consider the case where $\boldsymbol{x}$ and $\boldsymbol{w}$ have known signs and are sparse with respect to known dictionaries of size $K$ and $N$, respectively. Here, $K$ and $N$ may be larger than, smaller than, or equal to $L$. We introduce $\ell_1$-BranchHull, which is a convex program posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved. Under the assumptions that $\boldsymbol{x}$ and $\boldsymbol{w}$ satisfy a comparable-effective-sparsity condition and are $S_1$- and $S_2$-sparse with respect to a random dictionary, we present a recovery guarantee in a noisy case. We show that $\ell_1$-BranchHull is robust to small dense noise with high probability if the number of measurements satisfy $L\geq\Omega\left((S_1+S_2)\log^{2}(K+N)\right)$. Numerical experiments show that the scaling constant in the theorem is not too large. We also introduce variants of $\ell_1$-BranchHull for the purposes of tolerating noise and outliers, and for the purpose of recovering piecewise constant signals. We provide an ADMM implementation of these variants and show they can extract piecewise constant behavior from real images.

The effect of the terminal penalty in Receding Horizon Control for a class of stabilization problems

The Receding Horizon Control (RHC) strategy consists in replacing an infinite-horizon stabilization problem by a sequence of finite-horizon optimal control problems, which are numerically more tractable. The dynamic programming principle ensures that if the finite-horizon problems are formulated with the exact value function as a terminal penalty function, then the RHC method generates an optimal control. This article deals with the case where the terminal cost function is chosen as a cut-off Taylor approximation of the value function. The main result is an error rate estimate for the control generated by such a method, when compared with the optimal control. The obtained estimate is of the same order as the employed Taylor approximation and decreases at an exponential rate with respect to the prediction horizon. To illustrate the methodology, the article focuses on a class of bilinear optimal control problems in infinite-dimensional Hilbert spaces.

Joint Video Caching and Processing for Multi-Bitrate Videos in Ultra-Dense HetNets

Caching popular videos at the edge has been confirmed as a promising way to support low-latency video transmission and alleviate the backhaul traffic burden. Meanwhile, mobile edge computing (MEC) has also been regarded as an effective solution to meet the 5G low-latency service requirements. In this article, we propose to fully utilize both the storage and computing resources at edge servers to support multiple bitrate video streaming. We design the video caching, processing, and user association models that aim to minimize the average retrieval latency of all users. This problem is modeled as a mixed-integer bilinear problem, which is NP-hard . We show that under practical constraints on storage, bandwidth, and processing capacity, the problem does not exhibit sub-modular property and the performance of a greedy algorithm may not be strictly guaranteed. To deal with this challenging problem, we decompose the original problem into a cache placement problem and a user-BS association problem, while still preserving the interplay between the two sub-problems. A linearization and rounding algorithm, including: (i) a greedy rounding proactively caching scheme and (ii) a random-rounding user-BS association scheme, is then proposed, with performance bounds derived. Extensive simulation results show that the proposed scheme can achieve a near-optimal performance under various storage, computing capacity, and downlink bandwidth settings.

Joint channel and Doppler estimation for OSDM underwater acoustic communications

Orthogonal signal-division multiplexing (OSDM) has recently emerged as a promising modulation scheme for underwater acoustic communications. Although providing more flexibility in system design, it suffers from a special interference structure over time-varying channels. To enable reliable OSDM equalization, we propose a joint channel impulse response and Doppler estimation algorithm in this paper. Compared to the existing method, which solves a nonconvex bilinear optimization problem iteratively in the frequency domain without a guarantee of global optimality, the proposed algorithm transforms the problem into a convex quadratic minimization in the time domain and can achieve a closed-form solution.

Controllability of some bilinear and semilinear parabolic problems

Abstract We present in this paper a survey of recent results on the controllability of the parabolic system governed by bilinear control. We first discuss the problem of global controllability which corresponds to the question of whether the solution of the system can be driven to a given state at a some finite time by means of a control. We give some results on the global controllability of bilinear and semilinear reaction-diffusion equations. After this we introduce the case of partial controllability with the control acting on a subregion of the domain. Illustrative examples are also provided.

Boundedness of the bilinear Bochner-Riesz means in the non-Banach triangle case

In this article, we investigate the boundedness of the bilinear Bochner-Riesz means B α B^{\alpha } in the non-Banach triangle case. Bernicot et al. studied the bilinear Bochner-Riesz problem for n ≥ 2 n\geq 2 . Jeong, Lee, and Vargas improved their results in the Banach triangle case. We shall improve their results in the non-Banach triangle case. Improvement is reflected in two aspects: our partition of the non-Banach triangle is simpler; we obtain lower smoothness indices α ( p 1 , p 2 ) \alpha (p_{1},p_{2}) for various cases apart from 1 ≤ p 1 = p 2 > 2 1\leq p_{1}=p_{2}>2 .

A sequential quadratic Hamiltonian scheme for solving non-smooth quantum control problems with sparsity

A sequential quadratic Hamiltonian (SQH) scheme is investigated for solving non-smooth quantum optimal control problems with control costs that promote sparsity. Quantum control problems are a representative class of bilinear control problems that are central in application in nanosciences, and in this context the use of sparsity promoting functionals are desirable to facilitate the implementation of optimal controls in the laboratory.In this framework, it is shown that the SQH scheme provides many advantages with respect to other state-of-the-art methods, since it allows to also consider non-convex and discontinuous costs, without the need of regularization techniques. A theoretical discussion and results of numerical experiments are presented that demonstrate the effectiveness of the SQH scheme in solving control problems governed by a quantum spin system.

An efficient wavefield inversion: Using a modified source function in the wave equation

The wavefield is often reconstructed through solving a wave equation corresponding to an active source and using our best knowledge of the medium to reproduce the data that we seek to fit the observed data as part of a process we call full-waveform inversion (FWI). Alternatively, the wavefield can be inverted and used to invert for the velocity model, in an extended optimization problem, in which we can relax the requirement that the wavefield satisfies the wave equation. In this case, the wavefield calculation, as in FWI, requires a matrix inversion (which depends on the velocity) at practically every iteration. Thus, we formulate a bilinear optimization problem with respect to the wavefield and a modified source function, as independent variables. We specifically recast the wave equation so that velocity perturbations are included in this modified source function (which includes secondary or contrast sources); thus, it represents the velocity perturbations implicitly. The optimization includes a measure of the wavefield’s fit to the data at the sensor locations and the wavefield, as well as the modified source function, compliance with a wave equation corresponding to the background model. This problem is, however, complex with an extended model space that is ill-posed, so we use an alternating-direction method to reduce the inversion to two subproblems for inverting each of the wavefields and extended source. On the other hand, the velocity perturbations can be extracted in a separate step via direct division (deconvolution). Because we avoid using gradient methods in extracting the velocity perturbations, we are less prone to crosstalk artifacts when we use simultaneous sources. We evaluate these features on a simple two-anomalies model and the modified Marmousi model.

Engineering effective Hamiltonians

In the field of quantum control, effective Hamiltonian engineering is a powerful tool that utilizes perturbation theory to mitigate or enhance the effect that a variation in the Hamiltonian has on the evolution of the system. Here, we provide a general framework for computing arbitrary time-dependent perturbation theory terms, as well as their gradients with respect to control variations, enabling the use of gradient methods for optimizing these terms. In particular, we show that effective Hamiltonian engineering is an instance of a bilinear control problem—the same general problem class as that of standard unitary design—and hence the same optimization algorithms apply. We demonstrate this method in various examples, including decoupling, recoupling, and robustness to control errors and stochastic errors. We also present a control engineering example that was used in experiment, demonstrating the practical feasibility of this approach.

A branch-and-cut algorithm for Mixed-Integer Bilinear Programming

In this paper, we consider the Mixed-Integer Bilinear Programming problem, a widely-used reformulation of the classical mixed-integer quadratic programming problem. For this problem we describe a branch-and-cut algorithm for its exact solution, based on a new family of intersection cuts derived from bilinear-specific disjunctions. We also introduce a new branching rule that is specifically designed for bilinear problems. We computationally analyze the behavior of the proposed algorithm on a large set of mixed-integer quadratic instances from the MINLPlib problem library. Our results show that our method, even without intersection cuts, is competitive with a state-of-the-art mixed-integer nonlinear solver. As to intersection cuts, their extensive use at each branching node tends to slow down the solver for most problems in our test bed, but they are extremely effective for some specific instances.

Convergence and superconvergence of variational discretization for parabolic bilinear optimization problems

In this paper, we investigate a variational discretization approximation of parabolic bilinear optimal control problems with control constraints. For the state and co-state variables, triangular linear finite element and difference methods are used for space and time discretization, respectively, superconvergence in H^{1}-norm between the numerical solutions and elliptic projections are derived. Although the control variable is not discrete directly, convergence of second order in L^{2}-norm is obtained. These theoretical results are confirmed by two numerical examples.

Bounded bilinear control of coupled first-order hyperbolic PDE and infinite dimensional ODE in the framework of PDEs with memory

In this work, we consider the problem of bounded bilinear tracking control of a system of coupled first-order hyperbolic partial differential equation (PDE) with an infinite dimensional ordinary differential equation (ODE). This coupled PDE-infinite ODE system can be viewed as a degenerate system of two coupled first-order hyperbolic PDEs, the velocity of the ODE part vanishing. First, we convert this PDE-infinite ODE system into a first-order hyperbolic PDE with memory and investigate the bounded bilinear control problem in this framework. We consider as manipulated variable the constrained wave propagation velocity, which makes the control problem bounded and bilinear, and we take the measurements at the boundaries. To account for the actuator's constraints, we develop conditions under which the bounded control law ensures stability and tracking performances. This leads to a specification of the state-space region that enforces the desired system's closed-loop behaviour. To overcome the lack of full-state measurements, we design an observer-based bounded output-feedback control law which guarantees the reference tracking and uniform asymptotic stability of the system in closed-loop. A strong motivation of our work is the control problem of the solar collector parabolic trough where the manipulated control variable (the pump volumetric flow rate) is bilinear with respect to the PDE-infinite ODE model, and the measurements are taken at the boundary (tube's outlet). Simulation results illustrate the efficiency of the proposed control strategy.

Lower complexity bounds of first-order methods for convex-concave bilinear saddle-point problems

On solving a convex-concave bilinear saddle-point problem (SPP), there have been many works studying the complexity results of first-order methods. These results are all about upper complexity bounds, which can determine at most how many efforts would guarantee a solution of desired accuracy. In this paper, we pursue the opposite direction by deriving lower complexity bounds of first-order methods on large-scale SPPs. Our results apply to the methods whose iterates are in the linear span of past first-order information, as well as more general methods that produce their iterates in an arbitrary manner based on first-order information. We first work on the affinely constrained smooth convex optimization that is a special case of SPP. Different from gradient method on unconstrained problems, we show that first-order methods on affinely constrained problems generally cannot be accelerated from the known convergence rate $O(1/t)$ to $O(1/t^2)$, and in addition, $O(1/t)$ is optimal for convex problems. Moreover, we prove that for strongly convex problems, $O(1/t^2)$ is the best possible convergence rate, while it is known that gradient methods can have linear convergence on unconstrained problems. Then we extend these results to general SPPs. It turns out that our lower complexity bounds match with several established upper complexity bounds in the literature, and thus they are tight and indicate the optimality of several existing first-order methods.