Mathematics

We consider the problem of solving nonlinear optimization programs with stochastic objective and deterministic equality constraints. We assume for the objective that the function evaluation, the gradient, and the Hessian are inaccessible, while one can compute their stochastic estimates by, for example, subsampling. We propose a stochastic algorithm based on sequential quadratic programming (SQP) that uses a differentiable exact augmented Lagrangian as the merit function. To motivate our algorithm, we revisit an old SQP method \citep{Lucidi1990Recursive} developed for deterministic programs. We simplify that method and derive an adaptive SQP, which serves as the skeleton of our stochastic algorithm. Based on the derived algorithm, we then propose a non-adaptive SQP for optimizing stochastic objectives, where the gradient and the Hessian are replaced by stochastic estimates but the stepsize is deterministic and prespecified. Finally, we incorporate a recent stochastic line search procedure \citep{Paquette2020Stochastic} into our non-adaptive stochastic SQP to arrive at an adaptive stochastic SQP. To our knowledge, the proposed algorithm is the first stochastic SQP that allows a line search procedure and the first stochastic line search procedure that allows the constraints. The global convergence for all proposed SQP methods is established, while numerical experiments on nonlinear problems in the CUTEst test set demonstrate the superiority of the proposed algorithm.

Receding horizon optimal control problems compute the solution at each time step to operate the system on a near-optimal path. However, in many practical cases, the boundary conditions, such as external inputs, constraint equations, or the objective function, vary only marginally from one time step to the next. In this case, recomputing the optimal solution at each time represents a significant burden for real-time applications. This paper proposes a novel algorithm to approximately solve a perturbed constrained dynamic program that significantly improves the computational burden when the objective function and the constraints are perturbed slightly. The method hinges on determining closed-form expressions for first-order perturbations in the optimal strategy and the Lagrange multipliers of the perturbed constrained dynamic programming problem are obtained. This information can be used to initialize any algorithm (such as the method of Lagrange multipliers, or the augmented Lagrangian method) to solve the perturbed dynamic programming problem with minimal computational resources.

Deploying distributed renewable energy at the demand side is an important measure to implement a sustainable society. However, the massive small solar and wind generation units are beyond the control of a central operator. To encourage users to participate in energy management and reduce the dependence on dispatchable resources, a peer-to-peer energy sharing scheme is proposed which releases the flexibility at the demand side. Every user makes decision individually considering only local constraints; the microgrid operator announces the sharing prices subjective to the coupling constraints without knowing users' local constraints. This can help protect privacy. We prove that the proposed mechanism can achieve the same disutility and flexibility as centralized dispatch, and develop an effective modified best response based algorithm for reaching the market equilibrium. The concept of absorbable region is presented to measure the operating flexibility under the proposed energy sharing mechanism. A linear programming based polyhedral projection algorithm is developed to compute that region. Case studies validate the theoretical results and show that the proposed method is scalable.

The iteratively reweighted l1 algorithm is a widely used method for solving various regularization problems, which generally minimize a differentiable loss function combined with a nonconvex regularizer to induce sparsity in the solution. However, the convergence and the complexity of iteratively reweighted l1 algorithms is generally difficult to analyze, especially for non-Lipschitz differentiable regularizers such as nonconvex lp norm regularization. In this paper, we propose, analyze and test a reweighted l1 algorithm combined with the extrapolation technique under the assumption of Kurdyka-Lojasiewicz (KL) property on the objective. Unlike existing iteratively reweighted l1 algorithms with extrapolation, our method does not require the Lipschitz differentiability on the regularizers nor the smoothing parameters in the weights bounded away from zero. We show the proposed algorithm converges uniquely to a stationary point of the regularization problem and has local linear complexity--a much stronger result than existing ones. Our numerical experiments show the efficiency of our proposed method.

Optimization problems under affine constraints appear in various areas of machine learning. We consider the task of minimizing a smooth strongly convex function F(x) under the affine constraint Kx=b , with an oracle providing evaluations of the gradient of F and matrix-vector multiplications by K and its transpose. We provide lower bounds on the number of gradient computations and matrix-vector multiplications to achieve a given accuracy. Then we propose an accelerated primal--dual algorithm achieving these lower bounds. Our algorithm is the first optimal algorithm for this class of problems.

A quadratic binary unconstrained optimization model, hereafter QUBO, by definition is unconstrained. This, however, is not ideal if one needs to select a model containing only a fixed size binary vector. In this work we show how to add a constraint to a QUBO to force a particular size solution.

While the use of combination therapy is increasing in prevalence for cancer treatment, it is often difficult to predict the exact interactions between different treatment forms, and their synergistic/antagonistic effects on patient health and therapy outcome. In this research, a system of ordinary differential equations is constructed to model nonlinear dynamics between tumor cells, immune cells, and three forms of therapy: chemotherapy, immunotherapy, and radiotherapy. This model is then used to generate optimized combination therapy plans using optimal control theory. In-silico experiments are conducted to simulate the response of the patient model to various treatment plans. This is the first mathematical model in current literature to introduce radiotherapy as an option alongside immuno- and chemotherapy, permitting more flexible and effective treatment plans that reflect modern therapeutic approaches.

The ubiquitous expansion and transformation of the energy supply system involves large-scale power infrastructure construction projects. In the view of investments of more than a million dollars per kilometre, planning authorities aim to minimise the resistances posed by multiple stakeholders. Mathematical optimisation research offers efficient algorithms to compute globally optimal routes based on geographic input data. We propose a framework that utilizes a graph model where vertices represent possible locations of transmission towers, and edges are placed according to the feasible distance between neighbouring towers. In order to cope with the specific challenges arising in linear infrastructure layout, we first introduce a variant of the Bellman-Ford algorithm that efficiently computes the minimal-angle shortest path. Secondly, an iterative procedure is proposed that yields a locally optimal path at considerably lower memory requirements and runtime. Third, we discuss and analyse methods to output k diverse path alternatives. Experiments on real data show that compared to previous work, our approach reduces the resistances by more than 10% in feasible time, while at the same time offering much more flexibility and functionality. Our methods are demonstrated in a simple and intuitive graphical user interface, and an open-source package (LION) is available at this https URL.

Prolonged focused work periods decrease efficiency with related decline of attention and performance. Therefore, emergency fleet break scheduling should consider both area coverage by idle vehicles (related to the fleet's target arrival time to incidents) and vehicle crews' service requirements for breaks to avoid fatigue. In this paper, we propose a break assignment problem considering area coverage (BAPCAC) addressing this issue. Moreover, we formulate a mathematical model for the BAPCAC problem. Based on available historical spatio-temporal incident data and service requirements, the BAPCAC model can be used not only to dimension the size of the fleet at the tactical level, but also to decide upon the strategies related with break scheduling. Moreover, the model could be used to compute (suboptimal) locations for idle vehicles in each time period and arrange vehicles' crews' work breaks considering given break and coverage constraints.

This work 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 Babuska-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.