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In an earlier paper (https://doi.org/10.1137/21M1393315), the switch point algorithm was developed for solving optimal control problems whose solutions are either singular or bangbang or both singular and bangbang, and which possess a finite number of jump discontinuities in an optimal control at the points in time where the solution structure changes. The class of control problems that were considered had a given initial condition, but no terminal constraint. The theory is now extended to include problems with both initial and terminal constraints, a structure that often arises in boundaryvalue problems. Substantial changes to the theory are needed to handle this more general setting. Nonetheless, the derivative of the cost with respect to a switch point is again the jump in the Hamiltonian at the switch point.more » « lessFree, publiclyaccessible full text available December 1, 2024

The Polyhedral Active Set Algorithm (PASA) is designed to optimize a general nonlinear function over a polyhedron. Phase one of the algorithm is a nonmonotone gradient projection algorithm, while phase two is an active set algorithm that explores faces of the constraint polyhedron. A gradientbased implementation is presented, where a projected version of the conjugate gradient algorithm is employed in phase two. Asymptotically, only phase two is performed. Comparisons are given with IPOPT using polyhedralconstrained problems from CUTEst and the Maros/Meszaros quadratic programming test set.

Introduction Runners competing in races are looking to optimize their performance. In this paper, a runner's performance in a race, such as a marathon, is formulated as an optimal control problem where the controls are: the nutrition intake throughout the race and the propulsion force of the runner. As nutrition is an integral part of successfully running long distance races, it needs to be included in models of running strategies.
Methods We formulate a system of ordinary differential equations to represent the velocity, fat energy, glycogen energy, and nutrition for a runner competing in a longdistance race. The energy compartments represent the energy sources available in the runner's body. We allocate the energy source from which the runner draws, based on how fast the runner is moving. The food consumed during the race is a source term for the nutrition differential equation. With our model, we are investigating strategies to manage the nutrition and propulsion force in order to minimize the running time in a fixed distance race. This requires the solution of a nontrivial singular control problem.
Results As the goal of an optimal control model is to determine the optimal strategy, comparing our results against real data presents a challenge; however, in comparing our results to the world record for the marathon, our results differed by 0.4%, 31 seconds. Per each additional gel consumed, the runner is able to run 0.5 to 0.7 kilometers further in the same amount of time, resulting in a 7.75% increase in taking five 100 calorie gels vs no nutrition.
Discussion Our results confirm the belief that the most effective way to run a race is to run approximately the same pace the entire race without letting one's energies hit zero, by consuming inrace nutrition. While this model does not take all factors into account, we consider it a building block for future models, considering our novel energy representation, and inrace nutrition.

Abstract For polyhedral constrained optimization problems and a feasible point
, it is shown that the projection of the negative gradient on the tangent cone, denoted$$\textbf{x}$$ $x$ , has an orthogonal decomposition of the form$$\nabla _\varOmega f(\textbf{x})$$ ${\nabla}_{\Omega}f\left(x\right)$ . At a stationary point,$$\varvec{\beta }(\textbf{x}) + \varvec{\varphi }(\textbf{x})$$ $\beta \left(x\right)+\phi \left(x\right)$ so$$\nabla _\varOmega f(\textbf{x}) = \textbf{0}$$ ${\nabla}_{\Omega}f\left(x\right)=0$ reflects the distance to a stationary point. Away from a stationary point,$$\Vert \nabla _\varOmega f(\textbf{x})\Vert $$ $\Vert {\nabla}_{\Omega}f\left(x\right)\Vert $ and$$\Vert \varvec{\beta }(\textbf{x})\Vert $$ $\Vert \beta \left(x\right)\Vert $ measure different aspects of optimality since$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$ $\Vert \phi \left(x\right)\Vert $ only vanishes when the KKT multipliers at$$\varvec{\beta }(\textbf{x})$$ $\beta \left(x\right)$ have the correct sign, while$$\textbf{x}$$ $x$ only vanishes when$$\varvec{\varphi }(\textbf{x})$$ $\phi \left(x\right)$ is a stationary point in the active manifold. As an application of the theory, an active set algorithm is developed for convex quadratic programs which adapts the flow of the algorithm based on a comparison between$$\textbf{x}$$ $x$ and$$\Vert \varvec{\beta }(\textbf{x})\Vert $$ $\Vert \beta \left(x\right)\Vert $ .$$\Vert \varvec{\varphi }(\textbf{x})\Vert $$ $\Vert \phi \left(x\right)\Vert $ 
null (Ed.)A new method is developed for solving optimal control problems whose solutions are nonsmooth. The method developed in this paper employs a modified form of the Legendre–Gauss–Radau orthogonal direct collocation method. This modified Legendre–Gauss–Radau method adds two variables and two constraints at the end of a mesh interval when compared with a previously developed standard Legendre– Gauss–Radau collocation method. The two additional variables are the time at the interface between two mesh intervals and the control at the end of each mesh inter val. The two additional constraints are a collocation condition for those differential equations that depend upon the control and an inequality constraint on the control at the endpoint of each mesh interval. The additional constraints modify the search space of the nonlinear programming problem such that an accurate approximation to the location of the nonsmoothness is obtained. The transformed adjoint system of the modified Legendre–Gauss–Radau method is then developed. Using this transformed adjoint system, a method is developed to transform the Lagrange multipliers of the nonlinear programming problem to the costate of the optimal control problem. Fur thermore, it is shown that the costate estimate satisfies one of the Weierstrass–Erdmann optimality conditions. Finally, the method developed in this paper is demonstrated on an example whose solution is nonsmooth.more » « less