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We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simulations comparing entropic mirror descent and standard subgradient methods for the robust regression problem. √k) for strongly convex or convex objective functions, respectively, [15], [17].

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Current greenhouse gas emissions suggest that keeping global temperature increase below 1.5 degrees, as espoused in the Paris Agreements will be challenging, and to do so, the achievement of carbon neutrality is of utmost importance. It is also clear that no single solution can meet the carbon neutral challenge, so it is essential for scientific research to cover a broad range of technologies and initiatives which will enable the realization of a carbon free energy system. This study details the broad, yet targeted research themes being pioneered within the International Institute for Carbon-Neutral Energy Research (I2CNER). These approaches include hydrogen materials, bio-mimetic catalysts, electrochemistry, thermal energy and absorption, carbon capture, storage and management and refrigerants. Here we outline the state of the art for this suite of technologies and detail how their deployment, alongside prudent energy policy implementation can engender a carbon neutral Japan by 2050. Recognizing that just as no single technological solution will engender carbon neutrality, no single nation can expect to achieve this goal alone. This study represents a recognition of conducive international policy agendas and is representative of interdisciplinary, international collaboration.