Continuous-time Optimization Research Articles

Distributed Continuous-Time Algorithms for Time-Varying Constrained Convex Optimization

This paper is devoted to the distributed continuous-time optimization problems with time-varying objective functions and time-varying constraints. Different from most studied distributed optimization problems with time-invariant objective functions and constraints, the optimal solutions in this paper are time varying and form a trajectory. First, for the case where there exist only time-varying nonlinear inequality constraints, we present a distributed control algorithm that consists of a sliding-mode consensus part and a Hessian-based optimization part coupled with the log-barrier penalty functions. The algorithm can guarantee the asymptotical tracking of the optimal solution with a zero tracking error. Second, we extend the previous result to the case where there exist not only time-varying nonlinear inequality constraints but also linear equality constraints. An extended algorithm is presented, where quadratic penalty functions are introduced to account for the equality constraints and an adaptive control gain is designed to remove the restriction on knowing the upper bounds on certain information. The asymptotical convergence of the extended algorithm to the vicinity of the optimal solution is studied under suitable assumptions. The effectiveness of the proposed algorithms is illustrated in simulation. In addition, one proposed algorithm is applied to a multi-robot multi-target navigation problem with experimental demonstration on a multi-crazyflie platform to validate the theoretical results.

Predictive maintenance scheduling for multiple power equipment based on data-driven fault prediction

In view of the maintenance of multiple power equipment operating in abnormal conditions in large-scale power plants, based on the prediction of fault state, a predictive maintenance scheduling method is proposed to schedule the maintenance activities. Firstly, based on the actual operating data, combined with the influencing factors of fault state deterioration by Pareto analysis, a time-variant function is improved to predict the deterioration state of potential fault in future maintenance interval. Then, maintenance priority is divided based on the fault state, considering the constraints of maintenance resources and equipment downtime, with the objective of minimizing the total maintenance cost, a scheduling model is built for the maintenance of multiple equipment. Finally, aiming at the continuity of maintenance time, a two-stage algorithm is proposed, which divides the maintenance time-window to transform the complex continuous time optimization problem into a combinatorial optimization problem of time periods, and then develops the optimal maintenance scheme. Taking the maintenance of multiple power transformers as an example, combined with the data resources provided by Yunnan power grid of China, the effectiveness of the improved prediction function of fault state is proved. In addition, by comparing with the traditional maintenance strategy based on the principle of first-fault-first-repair, the superiority of the proposed maintenance scheduling method is verified in reducing cost and improving system stability.

Real-time Bidding for Time Constrained Impression Contracts in First and Second Price Auctions - Theory and Algorithms

We study the optimal bids and allocations in a real-time auction for heterogeneous items subject to the requirement that specified collections of items of given types be acquired within given time constraints. The problem is cast as a continuous time optimization problem that can, under certain weak assumptions, be reduced to a convex optimization problem. Focusing on the standard first and second price auctions, we first show, using convex duality, that the optimal (infinite dimensional) bidding policy can be represented by a single finite vector of so-called ''pseudo-bids''. Using this result we are able to show that the optimal solution in the second price case turns out to be a very simple piecewise constant function of time. This contrasts with the first price case that is more complicated. Despite the fact that the optimal solution for the first price auction is genuinely dynamic, we show that there remains a close connection between the two cases and that, empirically, there is almost no difference between optimal behavior in either setting. This suggests that it is adequate to bid in a first price auction as if it were in fact second price. Finally, we detail methods for implementing our bidding policies in practice with further numerical simulations illustrating the performance.

A piecewise conservative method for unconstrained convex optimization

We consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative force field generated by the objective function, without any kind of friction. We formulate a restart criterion based on the mean dissipation of the kinetic energy, and we prove a global convergence result for strongly-convex functions. Using the Symplectic Euler discretization scheme, we obtain an iterative optimization algorithm. We have considered a discrete mean dissipation restart scheme, but we have also introduced a new restart procedure based on ensuring at each iteration a decrease of the objective function greater than the one achieved by a step of the classical gradient method. For the discrete conservative algorithm, this last restart criterion is capable of guaranteeing a convergence result. We apply the same restart scheme to the Nesterov Accelerated Gradient (NAG-C), and we use this restarted NAG-C as benchmark in the numerical experiments. In the smooth convex problems considered, our method shows a faster convergence rate than the restarted NAG-C. We propose an extension of our discrete conservative algorithm to composite optimization: in the numerical tests involving non-strongly convex functions with $\ell^1$-regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.

Optimal investment strategies for pension funds with regulation-conform dynamic pension payment management in the absence of guarantees

In this article we consider the post-retirement phase optimization problem for a specific pension product in Germany that comes without guarantees. The continuous-time optimization problem is defined consisting of two specialties: first, we have a product-specific pension adjustment mechanism based on a certain capital coverage ratio which stipulates compulsory pension adjustments if the pension fund is underfunded or significantly overfunded. Second, due to the retiree’s fear of and aversion against pension reductions, we introduce a total wealth distribution to an investment portfolio and a buffer portfolio to lower the probability of future potential pension shortenings. The target functional in the optimization, that is to be maximized, is the client’s expected accumulated utility from the stochastic future pension cash flows. The optimization outcome is the optimal investment strategy in the proposed model. Due to the inherent complexity of the continuous-time framework, the discrete-time version of the optimization problem is considered and solved via the Bellman principle. In addition, for computational reasons, a policy function iteration algorithm is introduced to find a stationary solution to the problem in a computationally efficient and elegant fashion. A numerical case study on optimization and simulation completes the work with highlighting the benefits of the proposed model.

Distributed Nonconvex Optimization for Energy Efficiency in Mobile Ad Hoc Networks

This article proposes a distributed nonconvex optimization algorithm for energy efficiency (EE) in mobile ad hoc networks (MANETs). The proposed model for EE in MANETs can be classified into the category of nonconvex optimization problems. There are various distributed algorithms for solving nonconvex optimization problems. But, they are associated with some flaws such as complexity of implementation and low convergence rate. In this regard, in addition to the simplicity in implementation, the optimal point of the proposed model is a saddle point. Using augmented Lagrangian function with continuous-time optimization dynamics algorithm, a dynamic system will be established where its equilibrium points are Karush–Kuhn–Tucker points of the proposed model. Also, it will be shown that these points can be locally asymptotically stable under some conditions. Additionally, the proposed analytic method proves that the rate of convergence increases with an increasing penalty coefficient. To validate the performance of the presented algorithm, the problem of EE optimization for a sample mobile ad hoc network is simulated in MATLAB environment.

Cover’s Rebalancing Option with Discrete Hindsight Optimization

The author studies T. Cover’s rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to an initial deposit of 1 unit of the numéraire into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e., 200% equities and −100% bonds) and then continuously executing rebalancing trades so as to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalancing rules (i.e., 2) has some advantages over the pioneering approach taken by Cover &amp; Company in their theory of <i>universal portfolios</i> (1986, 1991, 1996, 1998), wherein one’s trading performance is benchmarked relative to the final wealth of the best unlevered rebalancing rule (of any kind) in hindsight. Our approach lets practitioners express an <i>a priori</i> view that one of the favored asset allocations (“bets”) in the set {<i>b</i>₁, …, <i>b_n</i>} will turn out to have performed spectacularly well in hindsight. In limiting our robustness to some discrete set of asset allocations (rather than all possible asset allocations), we reduce the price of the rebalancing option and guarantee that we will achieve a correspondingly higher percentage of the hindsight-optimized wealth at the end of the planning period. A practitioner who lives to delta-hedge this variant of Cover’s rebalancing option through several decades is guaranteed to see the day that his realized compound-annual capital growth rate is very close to that of the best <i>b_i</i> in hindsight, hence the point of the rock-bottom option price.

Trajectory Design via Convex Optimization for Six-Degree-of-Freedom Asteroid Powered Landing

This paper presents a convex programming algorithm based on modified Rodrigues parameters for a six-degree-of-freedom asteroid powered landing trajectory design problem. The trajectory design problem is formulated as a nonconvex fuel-optimal control problem with nonlinear coupled translational and rotational dynamics, nonconvex state and control constraints. In order to tackle the fuel-optimal control problem in the convexification framework, the original nonconvex continuous-time optimization problem is converted into a convex discrete-time optimization subproblem by linearizing and discretizing the dynamics and constraints. The effect of the coupling of translational and rotational dynamics on the convexification of nonconvex control constraints is discussed. The successive convexification method of solving a sequence of constrained convex subproblem is used to generate the optimal trajectory. The validity of the proposed algorithm for generating fuel-optimal trajectories and the effect of asteroid gravity on generated trajectories are examined through simulations of landing on different irregular asteroids. The Monte Carlo simulation is performed to examine the calculation performance of the proposed algorithm comparing to the Radau pseudospectral method algorithm and the proposed algorithm with the rotational motion parameterized by quaternions.

Distributed Time-Varying Optimization With State-Dependent Gains: Algorithms and Experiments

This brief addresses distributed continuous-time optimization problems with time-varying objective functions. The goal is for multiple agents to cooperatively minimize the sum of local time-varying objective functions with only local interaction and information while explicitly taking into account distributed adaptive gain design. Here, the optimal point is time varying and creates an optimal trajectory. First, for the unconstrained case, a distributed nonsmooth algorithm coupled with a state-dependent gain is proposed. It is shown that the interaction gain for each agent can be computed according to the variation of the Hessian and gradient information of the convex local objective functions so that the algorithm can solve the time-varying optimization problem without imposing a bound on any information about the local objective functions. Second, for the case where there exist common time-varying linear equality constraints, an extended algorithm is presented, where local Lagrangian functions are introduced to address the equality constraints. The asymptotic convergence of both algorithms to the optimal solution is proved. Numerical simulations are presented to illustrate the theoretical results. In addition, the one proposed algorithm is experimentally implemented and validated on a multi-Crazyflie platform.

Energy optimization through cooperative storage management: A calculus of variations approach

A framework to optimize energy utilization through battery management in a cooperative environment is proposed. Participating households share access to a community-owned energy farm and are equipped with lossy rechargeable batteries. The proposed optimization framework aims to minimize energy expenditure while accounting for time- and location-dependent electricity prices. The battery discharging schedules are optimized by using elements of calculus of variations and optimal control theory. The energy allocation policy is designed by solving an optimization problem through Lagrange Multipliers. Continuous-time optimization techniques are used to provide explicit solutions to the mathematical problems and determine closed-form performance estimates. The derived analytical expressions can be used to reduce the computational complexity of existing strategies, size energy storage systems, estimate performance bounds, and gain relevant insights. Extensive simulations are presented to validate the theoretical analysis and evaluate the impact of different system parameters. Specifically, cost savings are assessed as a function of discharging efficiency rates and nominal output power. It is shown that higher efficiency rates in battery operations lead to an increased price-selectivity of the proposed algorithm.

A new numerical method to solve pantograph delay differential equations with convergence analysis

The main aim presented in this article is to provide an efficient transferred Legendre pseudospectral method for solving pantograph delay differential equations. At the first step, we transform the problem into a continuous-time optimization problem and then utilize a transferred Legendre pseudospectral method to discretize the problem. By solving this discrete problem, we can attain the pointwise and continuous estimated solutions for the major pantograph delay differential equation. The convergence of method has been considered. Also, numerical experiments are described to show the performance and precision of the presented technique. Moreover, the obtained results are compared with those from other techniques.

Portfolio optimization for cointelated pairs: SDEs vs Machine learning

With the recent rise of Machine Learning (ML) as a candidate to partially replace classic Financial Mathematics (FM) methodologies, we investigate the performances of both in solving the problem of dynamic portfolio optimization in continuous-time, finite-horizon setting for a portfolio of two assets that are intertwined. In the Financial Mathematics approach we model the asset prices not via the common approaches used in pairs trading such as a high correlation or cointegration, but with the cointelation model in Mahdavi-Damghani (2013) that aims to reconcile both short-term risk and long-term equilibrium. We maximize the overall P&amp;L with Financial Mathematics approach that dynamically switches between a mean-variance optimal strategy and a power utility maximizing strategy. We use a stochastic control formulation of the problem of power utility maximization and solve numerically the resulting HJB equation with the Deep Galerkin method introduced in Sirignano and Spiliopoulos (2018). We turn to Machine Learning for the same P&amp;L maximization problem and use clustering analysis to devise bands, combined with in-band optimization. Although this approach is model agnostic, results obtained with data simulated from the same cointelation model gives a slight competitive advantage to the ML over the FM methodology1.

Continuous-Time Day-Ahead Operation of Multienergy Systems

Multienergy systems (MESs), as coupling of various energy sectors, can offer appropriate solutions to the operation of energy systems. The flexibility introduced by energy diversity can leverage the operation of MESs in supplying different energy demands, e.g., electricity, heat, etc. This article presents a continuous-time optimization framework for the day-ahead operation of MESs, where a function space solution method is proposed to reduce the dimensionality of the proposed model. The proposed methodology projects the day-ahead model into a finite-dimensional function space spanned by Bernstein polynomials and converts the problem with an indefinite dimension into a finite-order mixed-integer linear programming problem that can be easily solved with commercial solvers. The main purpose of the proposed framework is to model the parameter and decision trajectories of MESs in a way to reduce the day-ahead energy load approximation errors and minimize the costs associated with net-load deviations in real-time operation. In this way, the flexibility introduced by the operation of different energy units in MESs can be efficiently captured to further improve the performance of such systems. The proposed optimization framework is studied regarding four test MESs and the results for multiple cases are presented and compared. The results demonstrate the efficiency of the proposed methodology in capturing the load approximation errors, as it gives a better solution to the cost-efficient operation of MESs at a reasonable computational time.

Continuous-Time Optimization Dynamics Mirroring ADMM Architecture and Passivity-Based Robustification Against Delays

This article addresses a convex optimization problem that suffers from communication delays between a central collector and subsystems. First, we introduce continuous-time optimization dynamics that mirror the alternating direction method of multipliers architecture. Furthermore, we reveal that the proposed optimization dynamics have an architecture similar to that of bilateral teleoperation systems. By virtue of this passivity-based perspective, we apply a scattering transformation, which compensates for communication delays. Finally, we demonstrate the effectiveness of the proposed dynamics through numerical simulations of a grasp force optimization problem.

Hydrothermal scheduling in the continuous-time framework

Continuous-time optimization models have successfully been used to capture the impact of ramping limitations in power systems. In this paper, the continuous-time framework is adapted to model flexible hydropower resources interacting with slow-ramping thermal generators to minimize the hydrothermal system cost of operation. To accurately represent the non-linear hydropower production function with forbidden production zones, binary variables must be used when linearizing the discharge variables and the continuity constraints on individual hydropower units must be relaxed. To demonstrate the performance of the proposed continuous-time hydrothermal model, a small-scale case study of a hydropower area connected to a thermal area through a controllable high-voltage direct current (HVDC) cable is presented. Results show how the flexibility of the hydropower can reduce the need for ramping by thermal units triggered by intermittent renewable power generation. A reduction of 34% of the structural imbalances in the system is achieved by using the continuous-time model.

Distributed Second-Order Continuous-Time Optimization via Adaptive Algorithm with Nonuniform Gradient Gains

This paper addresses the distributed adaptive optimization problem over second-order multi-agent networks (MANs) with nonuniform gradient gains. A general convex function consisting of a sum of local differentiable convex functions is chosen as the team objective function. First, based on the local information of each agent’s neighborhood, a novel distributed adaptive optimization algorithm with nonuniform gradient gains is designed, where these gains only have relations with agents’ own states. And then, the original closed-loop system is changed into an equivalent one by taking a coordination transformation. Moreover, it is proved that the states including positions and velocities of all agents are bounded by constructing a Lyapunov function provided that the initial values are given. By the theory of Lyapunov stability, it is shown that all agents can finally reach an agreement and their position states converge to the optimal solution of the team objective function asymptotically. Finally, the effectiveness of the obtained theoretical results is demonstrated by several simulation examples.

Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

Continuous-time optimization is currently an active field of research in optimization theory; prior work in this area has yielded useful insights and elegant methods for proving stability and convergence properties of the continuous-time optimization algorithms. This article proposes novel gradient-flow schemes that yield convergence to the optimal point of a convex optimization problem within a fixed time from any given initial condition for unconstrained optimization, constrained optimization, and min-max problems. It is shown that the solution of the modified gradient-flow dynamics exists and is unique under certain regularity conditions on the objective function, while fixed-time convergence to the optimal point is shown via Lyapunov-based analysis. The application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient dominance, a relaxation of strong convexity. Then, a modified Newton's method is presented that exhibits fixed-time convergence under some mild conditions on the objective function. Building upon this method, a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed. Finally, the general min-max problem is considered, and a modified saddle-point dynamics to obtain the optimal solution in fixed time is developed.

Optimal Proportional Reinsurance Strategy Using Dynamic Programming

This paper is concerned with stochastic optimization in continuous time and its application in reinsurance. It deals with a model of optimization reinsurance which makes it possible to maximize the technical benefit of an insurance company and to minimize the risk for a given period. Indeed, here we discuss a stochastic optimal control problem to determine the optimal risk management strategy, the purpose of which is to identify the appropriate level of transfer rate for "quota share" reinsurance contracts, which are most appropriate to the risk profile of an insurance company. In addition, we use a resolution approach based on dynamic programming and the HJB equation in order to reach the aforementioned aim, which is finding the optimal strategy for reinsurance to adapt better for insurance company.

Time-varying continuous-time optimisation with pre-defined finite-time stability

In this paper, we propose a new family of continuous-time optimisation algorithms based on discontinuous second-order gradient optimisation flows, with finite-time convergence guarantees to local optima, for locally strongly convex (time-varying) cost functions. To analyse our flows, we first extend a well-know Lyapunov inequality condition for finite-time stability, to the case of (time-varying) differential inclusions. We then prove the convergence of these second-order flows in finite-time. In some particular cases, we can show that the finite-time convergence can be pre-defined by the user. We propose a robustification of the flows to bounded additive uncertainties and extend some of the results to the case of constrained optimisation. We show the performance of these flows on well-known optimisation benchmarks, namely, the Rosenbrock function, and the Rastringin function.

Timescale Separation in Autonomous Optimization

Autonomous optimization refers to the design of feedback controllers that steer a physical system to a steady state that solves a predefined, possibly constrained, optimization problem. As such, no exogenous control inputs such as set points or trajectories are required. Instead, these controllers are modeled after optimization algorithms that take the form of dynamical systems. The interconnection of this type of optimization dynamics with a physical system is however not guaranteed to be stable unless both dynamics act on sufficiently different timescales. In this paper, we quantify the required timescale separation and give prescriptions that can be directly used in the design of this type of feedback controllers. Using ideas from singular perturbation analysis, we derive stability bounds for different feedback laws that are based on common continuous-time optimization schemes. In particular, we consider gradient descent and its variations, including projected gradient, and Newton gradient. We further give stability bounds for momentum methods and saddle-point flows. Finally, we discuss how optimization algorithms such as subgradient and accelerated gradient descent, while well-behaved in offline settings, are unsuitable for autonomous optimization due to their general lack of robustness.