In an increasingly interconnected world, the efficient and economical transportation of individuals and commodities has emerged as a cornerstone of modern society. Optimizing transportation plans has a huge potential for journey planning, congestion reduction, supply chain management, and data exchanges. These strategies hold immense relevance not onlyin the realm of engineering and transportation, but also in other fields, such as physics, computer science, economics, and several subject areas in mathematics. The present thesis aims to elucidate the optimization of transportation strategies, with a particular focus on two classical problems, finding short paths in large networks and solving optimal transport problems with flow-rate constraints.

The so-called shortest path problem seeks an optimal path of transporting one unit of mass between pairs of vertices on graphs. We present a novel formulation of the problem as an l1-regularized regression, often referred to as lasso (Least Absolute Shrinkage and Selection Operator). Based on this formulation, we draw a connection specifically between trees that grow as active edge-sets in the least angle regression (LARS) algorithm of the lasso problem, and respective shortest-path trees that emerge using the bi-directional Dijkstra algorithm. Then, to overcome the dimensionality challenge in large graphs, we explore the alternating direction method of multipliers (ADMM) in the lasso formulation. The resulting derivativeproximal algorithm speeds up the search for the short paths, trading off optimality (i.e., finding shortest paths) that may not be absolutely essential in a variety applications.

The basic transport problem is motivated by the need to transport resources/mass between end-point distributions (supply and demand). We consider the classical Monge-Kantorovich optimal transport problem with a quadratic cost functional to penalize distance of transport, with an added constraint that transported mass is required to pass through constriction points while abiding by specified allowable flow-rate; constriction points may be conceptualized as toll stations with limited throughput. Our contributions in this topic are as follows: (1) we provide a precise Monge formulation for the optimal transport problem with flux constraint at constriction sites along the path that is amenable to generalization in higher dimensions. We work out in detail the case of transport in one dimension by proving existence and uniqueness of solutions. Under suitable regularity assumptions we give an explicit construction of the transport plan; (2) we provide a Kantorovich-type reformulation of the problem by introducing a marginal probability density for the time that mass-elements cross toll stations –a probability density that is to be determined so as to meet given flow-rate constraints. Interestingly, the Kantorovich-type formalism leads to multi-marginal optimal transport problem that is readily solvable by using linear programming. Moreover, existence and uniqueness of solutions are also established in this setting. Then, (3) we propose an entropic penalty term to regularize and reduce the computational cost of resulting multi-marginal problems. Entropic regularization of standard optimal transport leads to an efficient algorithm, the Sinkhorn algorithm, which applies in the present case as well. Leveraging the splittable nature of the cost in our formulation, we proposed a Gluing Sinkhorn algorithm for the multi-marginal optimal transport problem, which reduces the computational cost to a level comparable to that in standard two-marginal problems.

Classical thermodynamics is aimed at quantifying the efficiency of thermodynamic engines by bounding the maximal amount of mechanical energy produced compared to the amount of heat required under quasi-static operation. While this was accomplished early on, by Carnot and Clausius, the more practical problem of quantifying limits of output power remained elusive due to the fact that quasi-static processes require infinitely slow cycling, resulting in a vanishing power. Recent insights, drawn from stochastic thermodynamics, bridge the gap between theory and practice by presenting fresh approaches that lead to general laws applicable to the non-equilibrium system. Remarkably, the problem of minimizing dissipation over a finite time window can be expressed as a stochastic control problem leading to physically meaningful expressions for the dissipation cost in thermodynamic engines.

Building on the framework of stochastic thermodynamics, we derive bounds on the maximal power that can be drawn by cycling an overdamped ensemble of particles via a time-varying potential while alternating contact with heat baths of different temperature (Tc cold, and Th hot). The previous work focused on the setting where the potential is quadratic andthe distributions are Gaussian. Our analysis relaxes this assumption and asks about fundamental bounds on the power under arbitrary distribution. Specifically, first by casting the optimization problem of the power into the Benamou-Brenier formulated Optimal mass transport problem, we show that the power output is bounded by the Fisher information of the boundary distribution, which can become unbounded with complicated and irregular control. However, it is unreasonable to expect technological solutions to such demands, and therefore, a constraint on the complexity of the potential seems meaningful. Assuming a suitable bound M on the spatial gradient of the controlling potential, we show that the maximal achievable power is bounded by M/8(Th/Tc-1). Moreover, we show that this bound can be reached to within a factor of (Th/Tc-1)/(Th/Tc+1) by operating the cyclic thermodynamic process with a quadratic potential.

While the majority of works in the literature are concerned with thermodynamic engines operating in the setting of Carnot's cyclic contact with alternating heat baths, the natural processes in living organisms do not follow this setting. Instead, it is the periodic fluctuations in chemical concentrations in conjunction with the variability of electrochemical potentials that provide the universal source of cellular energy. Thus, energy exchange is often mediated by continuous processes and energy differentials, whereas the Carnot cycle reflects the switching mechanics of an idealized engine. We herein propose to consider thermodynamic processes driven by a heat bath with periodic and continuous temperature profile, and study questions of maximal power and efficiency at maximal power. Our results state that themaximal power satisfies a bound proportional to the average fluctuations in the temperature profile. Moreover, we show the surprising result that the efficiency at maximal power does not depend on the specific temperature profile, only the maximum and minimum.