Reliable wind power generation forecasting is crucial to meet energy demand, to trade and invest. We propose a model to simulate and quantify uncertainties in such forecasts. This model is based on Stochastic Differential Equations whose time-dependent parameters are inferred using continuous optimization of an approximate Likelihood function. The result is a skew stochastic process that simulates uncertainty of wind power forecasts accounting for maximum power production limit and other temporal effects. We apply the model to historical Uruguayan data and forecasts.
Adam is a popular variant of the stochastic gradient descent. However, only few information about its convergence is available in the literature. We investigate the dynamical behavior of Adam when the objective function is non-convex and differentiable. We introduce a continuous-time version of Adam, under the form of a non-autonomous ordinary differential equation. Existence, uniqueness and convergence of the solution to the stationary points of the objective function are established. It is also proved that the continuous-time system is a relevant approximation of the Adam iterates.
Motivated by a contact model in biomedicine, a compliant obstacle problem arises exemplarily in the situation of two elastic membranes enclosing a region of constant volume, which are subject to external forces. The motivating model enables a reformulation as a obstacle-type quasi-variational inequality (QVI) with volume constraints. We analyse this class of QVIs and present a tailored path-following semismooth Newton method to solve the problem. In view of the numerical performance, adaptive finite elements are used based on a posteriori error estimation.
Uruguay has always been a pioneer in the use of renewable sources of energy. Nowadays, it can usually satisfy its total demand from renewable sources, but half of the installed power, due to wind and solar sources, is non-controllable and has high uncertainty and variability. We deal with non-Markovian dynamics through a Lagrangian relaxation, solving then a sequence of HJB PDEs associated with the system to find time-continuous optimal control and cost function. We introduce a monotone scheme to avoid spurious oscillations. Finally, we study the usefulness of extra system storage capacity.
A robot that works no matter how well designed, the results of its operation in the actual environment are sometimes quite different from the expected results of its design environment. In this study, we train a snake-type robot Snaky with 8 joints and 4 control rotary motors to learn to complete the specified snake mission through an autonomous learning mechanism. The core learning mechanism is the artificial neuromolecular system (ANM) developed by the research team in the early days. The learning continues until the robot achieves the assigned tasks or stopped by the system developer.
The paper presents a simple technique to heuristically reduce the column set of set covering problems. The idea consists in discarding all columns that are not selected by the subgradient algorithm used to solve the Lagrangian relaxation. In case the reduction leaves a large number of columns, this filter may be strengthened by discarding all columns that are not selected more than alpha% times.the proposed technique is in the class of "Sifting" techniques. We are evaluating this technique using a Benders algorithm designed for solving real world set covering instances arising from Deutsche Bahn.
First-order methods for linear feasibility problems have been known since the Kaczmarz algorithm. According to Naumann and Klee, any polytope is obtainable as a section of higher-dimensional cube. Transformation of linear constraints to such form is very much like the one in simplex method, but instead of non-negativity constraints every variable is constrained to an interval. A solution of LP problem amounts to finding an intersection between a line and an affine transformation of the cube, which iteratively uses calculation of a hyperplane separating a point on the line and that polytope.
This poster presents recent advances in dolfin-adjoint, the algorithmic differentiation tool for the finite element framework FEniCS. We will show dolfin-adjoint can be used to differentiated and optimised PDE models in FEniCS that are coupled to neural network models implemented in pyTorch. Our approach allows to compute derivatives with respect to weights in the neural network, and as PDE coefficients such as initial condition, boundary conditions and even the mesh shape.
Within model predictive control (MPC), an infinite time horizon problem is split into a sequence of open-loop finite horizon problems which allows to react to changes in the problem setting due to external influences. In order to find suitable time horizon lengths, we utilize a residual-based time adaptive approach which uses a reformulation of the optimality system of the open-loop problem as a biharmonic equation. In order to achieve an efficient numerical solution, we apply POD based model order reduction.
The absolute value equation plays a outstanding role in optimization. In this paper, we employ proximal gradient method to present a efficient solution for solving it. We use accelerated Bregman proximal gradient methods that employ the Bregman distance of the reference function as the proximity measure.
We propose a symbolic-numeric method to solve a constrained optimization problem with free parameters where all the functions are polynomials. The proposed method is based on the quadratic penalty function method and symbolically performs the limit operation of a penalty parameter by using projective spaces and tangent cones. As its output, we obtain an implicit function representation of optimizers as a function of the free parameters. Some examples show that the exact limit operation enables us to find solutions that do not satisfy the Karush-Kuhn-Tucker (KKT) conditions.
We consider an advanced queueing model with energy harvesting and multi-threshold control by the service modes. The account of the possibility of error occurrence is very important in modelling wireless sensor networks. This model generalizes a previously considered in the literature by suggestions that the buffer for customers has an infinite capacity and the accumulated energy can leak from the storage. Under the fixed thresholds defining the control strategy, behavior of the system is described by multi-dimensional Markov chain what allows to formulate and solve optimization problems.
We show how a nonlinear scalarization functional can be used in order to characterize several well-known set relations and which thus plays a key role in set optimization. By means of this functional, we derive characterizations for minimal elements of set-valued optimization problems using a set approach. Our methods do not rely on any convexity assumptions on the considered sets. Furthermore, we introduce a derivative-free descent method for set optimization problems without convexity assumptions to verify the usefulness of our results.
We show the construction of a novel set relation implementing a compromise between the known concepts "upper set less relation" on the one hand and "lower set less relation" on the other. The basis of the construction lies in the characterization of the relation of two sets by means of a single real value that can be computed using nonlinear separation functionals of Tammer-Weidner-type. The new relation can be used for the definition of flexible robustness concepts and the theoretical study of set optimization problems based on this concepts can be based on properties of the other relations.
Resistance spot welding gun is generally used to bond parts in the automotive and home appliance industry. The industrial robot is essential because most of the welding process is automated. The weight of the welding gun should be minimized to increase the efficiency of the robot motion. In this study, welding gun frame optimization is performed using genetic algorithm methods. Several parts of frame thickness are specified as variables and the optimization model is validated through both simulation and experiments. The weight is decreased by up to 14% and cost saving is also achieved.
We consider a periodic-review pricing problem. There are firms i=1,...,N competing in the market of a commodity in a stationary demand environment. Each firm can modify the price over discretized time. The firm i's demand at period n consists of the linear model conditional on the selling prices of its own and other firms with an independent and identically distributed random noise. The coefficients of the linear demand model is not known a priori. Each firms' objective is to sequentially set prices to maximize revenues under demand uncertainty and competition.
In order to efficiently design experiments, decision makers have to compare them. The comparison of experiments due to Blackwell is based on the notion of garbling. It is well-known that the Blackwell order can be characterized by agents’ maximal expected utility. We focus on the comparison of experiments due to Shannon by allowing pre-garbling. We show that for any subset of decision makers the comparison of their maximal expected utilities does not capture the Shannon order. For characterizing the Shannon order, we enable decision makers to redistribute noise, and convexify the garbling.
We present SVAG, a variance reduced stochastic gradient method with SAG and SAGA as special cases. A parameter interpolates and extrapolates SAG/SAGA such that the variance the gradient approximation can be continuously adjusted, at the cost of introducing bias. The intuition behind this bias/variance trade-off is explored with numerical examples. The algorithm is analyzed under smoothness and convexity assumptions and convergence is established for all choices of the bias/variance trade-off parameter. This is the first result that simultaneously captures SAG and SAGA.
We are interested in solving an online learning problem when the decision variable is a function $f$ which belongs to reproducing kernel Hilbert space. We present the dynamic regret analysis for an optimally compressed online function learning algorithm. The dynamic regret is bounded in terms of the loss function variation. We also present the dynamic regret analysis in terms of the function space path length. We prove that the learned function via the proposed algorithm has a finite memory requirement. The theoretical results are corroborated through simulation on real data sets.
Typical cyber-physical Systems consist of many interconnected heterogeneous entities that share resources such as communication or computation. Here optimal control and resource scheduling go hand in hand. In this work, we consider the problem of simultaneous control and scheduling. Specically, we present DIRA, a Deep Reinforcement Learning based Iterative Resource Allocation algorithm, which is scalable and control sensitive. For the controller, we present three compatible designs based on actor-critic reinforcement learning, adaptive linear quadratic regulation and model predictive control.
We present a modified version of spectral partitioning utilizing the graph p-Laplacian, a nonlinear variant of the standard graph Laplacian. The computation of the second eigenvector is achieved with the minimization of the graph p-Laplacian Rayleigh quotient, by means of an accelerated gradient descent approach. A continuation approach reduces the p-norm from a 2-norm towards a 1-norm, thus promoting sparser solution vectors, corresponding to smaller edgecuts. The effectiveness of the method is validated in a variety of complex graphical structures originating from power and social networks.
We consider a biobjective PDE constrained shape optimization problem maximizing the reliability of a ceramic component under tensile load and minimizing its material consumption. The numerical implementation is based on a finite element discretization with a B-spline representation of the shapes. Using the B-spline coefficients as control variables we combine it with a single objective gradient descent algorithm for solving the weighted sum scalarization and apply it to 2D test cases. While varying the weights we determine different solutions representing different preferences.
In this paper, a redundancy allocation problem (RAP) with an active strategy and choice of a component type is considered, in which there are different components with various reliabilities and costs considered as imprecise and varied in an interval uncertainty set. Therefore, a robust model for the RAP is presented to propose a robust design for different realization of uncertain parameters. Finally, uncertainty and price discounting are considered together in a comprehensive model for the RAP and the effects on warranty costs with minimal repair are investigated.
Bilinear systems are linear on phase coordinates when the control is fixed, and linear on the control when the coordinates are fixed. They describe the dynamic processes of nuclear reactors, kinetics of neutrons, and heat transfer. Further investigations show that many processes in engineering, biology, ecology and other areas can be described by the bilinear systems . The problem is to find optimal control minimizing the quadratic functional for the system driven by white noise with small parameter. Successive approximations are constructed by perturbation theory method/
In recent years, generalized Nash equilibrium problems in function spaces involving control of PDEs have gained increasing interest. One of the central issues arising is the question of existence, which requires the topological characterization of the set of minimizers for each player. In this talk, we propose conditions on the operator and the functional, that guarantee the reduced formulation to be a convex minimization problem. Subsequently, we generalize the results of convex analysis to derive optimality systems also for nonsmooth operators. Our findings are illustrated with examples.
The average vector field (AVF) discrete gradient, originally designed by Quispel and McLaren as an energy-preserving numerical integrator for Hamiltonian systems has found a variety of applications, including Optimization. In contrast to the very recent work of Ehrhardt et. al. which deals with AVF discretizations of gradient flows, the present contribution aims at deriving discrete versions of second-order gradient systems based on AVF discrete gradients. The obtained optimization algorithms are shown to inherit the asymptotic behavior of their continuous counterparts.
This paper concerns robust portfolio optimization problem which is a popular topic in financial engineering. To reach the investor’s requirement on asset selection, we build a sparse robust portfolio optimization model by adding a nonconvex penalty item into the objective function instead of the cardinality constraint. A series of models based on different uncertainty sets for expected return rates and covariance are present. The Alternating Quadratic Penalty (AQP) method is applied to solve the relative sparse robust portfolio selection models.
First, we consider a class of nonconvex optimization problems including l0 norm minimization and dictionary learning which can be efficiently solved using ADMM and its variants. In particular, the nonconvex problem is decomposed such that each subproblem is either a tractable convex program or endowed with hidden convexity. Competitive performance is demonstrated through preliminary numerics. Second, we show that for a general class of nonconvex optimization problems, proximal ADMM is able to avoid strict saddle points with probability 1 using random initialization.
This paper considers a hybrid system deploying both wind and solar photovoltaic (PV) farms at the same site. We propose a simulation optimization approach. The simulation of uncertain wind, sky irradiance, and dust deposition is based on a meteorological dataset. The co-siting constraints which degrade the production are handled by calculating wind-turbine’s umbra and penumbra on PVs, planning shortest roads for routing the maintenance vehicles, and minimizing the land usage conflict by an evolutionary algorithm. We propose several models to leverage the production, cost, and uncertainty.
Motivated by practical challenges in controlling hybrid systems (i.e. robot trajectory optimization through contact), we propose a machine learning approach to speed up the mixed-integer optimization (MIO) for numerical optimal control. We learn policies to serve as a form of solution memory by casting the training as learning from suboptimal demonstrations. The resulting policies provide feasible incumbent solutions for the branch-and-bound (-cut) process of MIO. Coupled with other insights developed in this work, the approach speeds up numerical optimal control of hybrid systems.