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The dynamics of unmanned aerial systems (UAS) are often nonlinear, especially at high speed and high maneuverability flight. The inherent nonlinearity of the system renders conventional linear control techniques inadequate for achieving optimal control outcomes. Consequently, unmanned aerial vehicle (UAV) trajectory planning is fundamentally a nonlinear optimal control challenge, characterized by the difficulty of swiftly obtaining a stable solution and the propensity to converge on suboptimal local solutions during the solving process. In response to this, a method for autonomous UAV obstacle avoidance trajectory planning is introduced, leveraging a convex optimization-based particle swarm algorithm. The UAV uses sensors to record the information about obstacles, adopts a one-dimensional time-parameterized polynomial trajectory to construct the obstacle avoidance control input, combines the safety penalty function between the navigation path and obstacles to generate the obstacle avoidance trajectory planning objective function, and obtains the path points in the obstacle avoidance trajectory; the particle swarm algorithm is used to solve the objective function, the parameters such as obstacle avoidance control input and safety penalty function are used as particles, and the optimal parameter values are obtained through the iterative updating process of position and speed. Aiming at the acquired obstacle avoidance trajectory path points, the convex optimization algorithm is used to construct a non-convex optimal control model that meets the requirements of energy optimization, and is rooted in the concave-convex process, so that the control model’s objective function and inequality constraint are both convex, while the formula constraint is affine. By transforming the control model into a convex optimization problem, it aligns with the energy optimization requirements. The sequential convex optimization framework is employed to solve this problem, enabling the optimization of the UAV’s trajectory for obstacle avoidance. Experimental outcomes demonstrate that this approach effectively captures the coordinate details of obstacles, navigates through various dense obstacle scenarios, and simultaneously guarantees an energy-efficient path to the target destination.

We propose an efficient algorithm for computing the equilibrium of a capital asset pricing model with heterogeneous investors and short-sale constraints. We show that the equilibrium prices of the risky assets in the model are proportional to the Lagrangian multipliers of an equivalent dual formulation of the problem. Based on this observation, we derive sufficient conditions to guarantee the existence and uniqueness of equilibrium and prove the convergence of the algorithm. Numerical examples are also provided to illustrate the algorithm.

In this paper, we study convex semi-infinite programming involving minimax problems. One of the difficulties in solving these problems is that the maximum type functions are not differentiable. Due to the nonsmooth nature of the problem, we apply the special proximal bundle scheme on the basis of ${𝒱}{𝒰}$-decomposition theory to solve the nonsmooth convex semi-infinite minimax problems. The proposed scheme requires an evaluation within some accuracy for all the components of the objective function. Regarding the incremental method, we only need one component function value and one subgradient which are estimated to update the bundle information and produce the search direction. Under some mild assumptions, we present global convergence and local superlinear convergence of the proposed bundle method. Numerical results of several example problems are reported to show the effectiveness of the new scheme.

In many high energy experiments, the physics quantities are obtained by measuring the cross-sections at a few energy points over an energy region. This was referred to as scan experiment. The optimal design of the scan experiment (how many energy points, what the energies are, and what is the luminosity at each energy point) is of great significance both for scientific research and from economical viewpoint. Two approaches, one has recourse to the sampling technique and the other resorts to the analytical proof, are adopted to figure out the optimized scan scheme for the relevant parameters. The final results indicate that for $n$ parameters scan experiment, $n$ energy points are necessary and sufficient for optimal determination of these $n$ parameters; each optimal position can be acquired by single parameter scan (sampling method), or by analysis of auxiliary function (analytic method); the luminosity allocation among the points can be determined analytically with respect to the relative importance between parameters. By virtue of the second optimization theory established in this paper, it is feasible to accommodate the perfectly optimal scheme for any scan experiment.

This note proposes a systematic method to optimizing dynamic output feedback controller for continuous-time hybrid systems in the Piecewise Affine (PWA) form. In this note, design of the dynamic output feedback controller is shown in the frame of L₂ gain synthesis for PWA systems. Matrix inequalities derived from dynamic output feedback controller is not usually linear. Using a congruence transformation, the nonlinear matrix inequalities are mapped into linear matrix inequalities, whose solutions are quite feasible using commercially available software. The proposed techniques satisfy features such as: globally quadratic stability, continuity of control signal, and tracking a reference input. A numerical example is included to demonstrate the applicability and effectiveness of the proposed technique.

With advancements in computing and communication technologies on mobile devices, the performance requirements of embedded processors have significantly increased, resulting in a corresponding increase in its energy consumption. Dynamic scaling of operating voltage and operating frequency has a strong correlation to energy minimization in CMOS real-time circuits. Simultaneous optimization of ($v$, $f)$ pairs under dynamic activity levels is thus extensively investigated over several years. The supply voltage is tuned dynamically during runtime (DVS), with a fixed threshold voltage, to achieve energy minimization. This work addresses the issue of maximizing the energy efficiency of real-time periodic, aperiodic and mixed task sets, in a uniprocessor system, by developing a novel task feasibility methodology, with a novel processor performance-based constraint, to generate the optimal operating supply voltage to the individual task of task sets. The energy minimization of real-time mixed task sets is formulated as Geometric Programming (GP) problem, by varying frequency for periodic tasks sets and keeping fixed frequency for aperiodic tasks set, over a range of task sets and hence computing optimal operating voltages. Simulation experiments show energy savings on the cumulative basis of 50%, 38% and 29% for periodic, aperiodic and mixed task sets, respectively, based on the processing timing constraints of task sets.

A privacy protection task offloading algorithm combining discrete differential evolution and convex optimization is proposed to resolve the new position privacy leakage problem for mobile terminal due to transferring tasks to multiple edge servers for computation via wireless networks. First, the privacy cost of offloading decision is defined for the characteristics of the position privacy problem combined with the standard deviation theory. Second, the objective problem is modeled as a joint optimization of position privacy protection, offloading decision and transmission power, and the total cost of task offloading is an optimization objective. Finally, to handle the objective problem efficiently, offloading decision variable is decoupled from the transmission power variable to decrease the computational complexity of the problem; the discrete differential evolution algorithm is designed to accumulate the approximate optimal offloading decision, and the optimal transmission power is efficiently solved by variable substitution method and subgradient method. Simulations reveal that the algorithm has excellent convergence; it improves the position privacy protection level of mobile terminal and reduces the total cost of task offloading while maintaining lower energy consumption and latency.

This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced.

In this paper, we describe a fast and efficient method for multi-modal and discontinuity-preserving image registration, implemented on graphics hardware. Multi-sensory data fusion and medical image analysis often pose the challenging task of aligning dense, non-rigid and multi-modal images. However, also optical sequences or stereo image pairs may present variable illumination conditions and noise. The above problems can be addressed by an invariant similarity measure, such as mutual information. Additionally, when using a regularized approach to deal with the ill-posedness of the problem, one has to take care of preserving discontinuities at the motion boundaries. Our approach efficiently addresses the above issues through a primal-dual convex estimation framework, using an approximated Hessian matrix that decouples pixel dependencies, while being asymptotically correct. At the same time, we achieve a high computational efficiency by means of pre-quantized kernel density estimation and differentiation, as well as a parallel implementation on the GPU. Our approach is demonstrated on ground-truth data from the Middlebury database, as well as medical and visible-infrared image pairs.

This paper proposes a numerical approach for computing bounds for the arbitrage-free prices of an option when some options are available for trading. Convex duality reveals a close relationship with recently proposed calibration techniques and implied trees. Our approach is intimately related to the uncertain volatility model of Avellaneda, Levy and Parás, but it is more general in that it is not based on any particular form of the asset price process and does not require the seller's price of an option to be a differentiable function of the cash-flows of the option. Numerical tests on S&P500 options demonstrate the accuracy and robustness of the proposed method.

We develop a model for indifference pricing in derivatives markets, where price quotes have bid–ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investor’s views, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimization allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between superhedging and subhedging prices.

We investigate various properties of the sublevel set G = {x : g(x) ≤ 1} and the integration of h on this sublevel set when g and h are positively homogeneous functions (and in particular homogeneous polynomials). For instance, the latter integral reduces to integrating hexp(-g) on the whole space ℝⁿ (a nonGaussian integral) and when g is a polynomial, then the volume of G is a convex function of the coefficients of g. We also provide a numerical approximation scheme to compute the volume of G or integrate h on G (or, equivalently to approximate the associated nonGaussian integral). We also show that finding the sublevel set {x : g(x) ≤ 1} of minimum volume that contains some given subset K is a (hard) convex optimization problem for which we also propose two convergent numerical schemes. Finally, we provide a Gaussian-like property of nonGaussian integrals for homogeneous polynomials that are sums of squares and critical points of a specific function.

A lot of works in machine learning (ML) usually use components which are built by signal processing, such as wavelet and Fourier coefficients, for computations; therefore, many problems in signal processing affect ML algorithms. Consequently, we try to solve linear inverse problems in signal processing, such as the classical denoising and deconvolution problems, in this work. The penalized least squares regression (PLSR), also called as the proximity operator, of the non-convex regularization, can solve various problems in signal processing. In the classical denoising problem, the PLSR with the double threshold value is more flexible than the PLSR with the single threshold value. Therefore, we propose the novel non-convex regularization which can build the PLSR with the double threshold value. The proposed regularization is based on good properties of the minimax-concave (MC) regularization. We also present novel PLSRs of the MC and proposed regularizations where these PLSRs have closed-form solutions and the relationship to the group of considered components, also known as the multivariate case, together. We compare proposed methods with state-of-the-art methods in the classical denoising and deconvolution problems. Here, we use the majorization-minimization (MM) method for comparing the performance of regularizations in the deconvolution problem. Experimental results show that proposed methods give trustworthy results.

In this paper, a new theory is developed for first-order stochastic convex optimization, showing that the global convergence rate is sufficiently quantified by a local growth rate of the objective function in a neighborhood of the optimal solutions. In particular, if the objective function $F(w)$ in the $𝜖$-sublevel set grows as fast as $\parallel w-w_{\ast }{\parallel }_2^{1/𝜃}$, where $w_{\ast }$ represents the closest optimal solution to $w$ and $𝜃\in (0,1]$ quantifies the local growth rate, the iteration complexity of first-order stochastic optimization for achieving an $𝜖$-optimal solution can be $\tilde{O}(1/{𝜖}^{2(1-𝜃)})$, which is optimal at most up to a logarithmic factor. To achieve the faster global convergence, we develop two different accelerated stochastic subgradient methods by iteratively solving the original problem approximately in a local region around a historical solution with the size of the local region gradually decreasing as the solution approaches the optimal set. Besides the theoretical improvements, this work also includes new contributions toward making the proposed algorithms practical: (i) we present practical variants of accelerated stochastic subgradient methods that can run without the knowledge of multiplicative growth constant and even the growth rate $𝜃$; (ii) we consider a broad family of problems in machine learning to demonstrate that the proposed algorithms enjoy faster convergence than traditional stochastic subgradient method. We also characterize the complexity of the proposed algorithms for ensuring the gradient is small without the smoothness assumption.

Implicit regularization refers to the property of optimization algorithms to be biased towards a certain class of solutions. This property is relevant to understand the behavior of modern machine learning algorithms as well as to design efficient computational methods. While the case where the bias is given by a Euclidean norm is well understood, implicit regularization schemes for more general classes of biases are much less studied. In this work, we consider the case where the bias is given by a strongly convex functional, in the context of linear models, and data possibly corrupted by noise. In particular, we propose and analyze accelerated optimization methods and highlight a trade-off between convergence speed and stability. Theoretical findings are complemented by an empirical analysis on high-dimensional inverse problems in machine learning and signal processing, showing excellent results compared to the state of the art.

In this paper, we propose an online state tomography method for $N$-qubit quantum system based on the continuous weak measurement and compressed sensing (CS). The quantum system is described by stochastic master equation. The continuous weak measurement operators for the $N$-qubit quantum system, which are indirectly acted on the quantum system, are derived according to the measurement operator results of two-level quantum system. The online time-varying measurement operators are obtained by means of the dynamic evolution equation of the system. The quantum state is online estimated by solving the optimization problem of minimizing the two-norm with the positive definite constraints of density matrix, and we use the nonnegative least squares algorithm to solve the optimization problem. CS theory is used to reduce the number of the measurements in the process of online state estimation. In the numerical experiments, we study the effects of external control field, measurement rate and the different numbers of qubits on the performance in the proposed method. The minimum required numbers of measurements for 2, 3, 4 and 5 qubits are found. The normalized distance and fidelity of our proposed method can achieve satisfying accuracy with small number of measurements.

We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.

In this paper, distributed optimization control for a group of autonomous Lagrangian systems is studied to achieve an optimization task with local cost functions. To solve the problem, two continuous-time distributed optimization algorithms are designed for multiple heterogeneous Lagrangian agents with uncertain parameters. The proposed algorithms are proved to be effective for those heterogeneous nonlinear agents to achieve the optimization solution in the semi-global sense, even with the exponential convergence rate. Moreover, simulation adequately illustrates the effectiveness of our optimization algorithms.

The trajectory planning and tracking problem are critical points of intelligent vehicles concerning their safety and stability. If these parts are separated, interactions should be made between them, especially when sudden changes and disturbances appear. This paper presents a method that integrates the two parts using a cascade structure. Additionally, the proposed method deals with the interaction between the lateral and the longitudinal trajectory based on dynamical considerations. The whole problem is handled using the model predictive method based on online optimization. This system receives the path borders as input and generates the control requests for the actuators on its output. The configuration space of the system can be maximized to gain stability by handling the lateral-longitudinal parts and the trajectory planning and tracking in one complex system. The main advantage of the proposed approach is that the optimization problem in the predictive method is formulated so that the path and dynamics are considered equality and inequality constraints, and the cost function includes only a physical phenomenon to be minimized without tuning parameters. The evaluation of the proposed algorithm is presented in this paper based on simulation and real-time measurements results.

In this paper, we provide the implications of using the performance ratio being defined by the expected shortfall-well known as RAROC-in static portfolio optimization, by giving the proof of the relevant results. We also use RAROC as a primary function in the AUGMECON algorithm, providing the main scheme of such an application on data from Athens Stock Exchange. The use of RAROC in AUGMECON as a primary function is also faced as an optimization problem under a general nonconvex optimization framework.