Journal of the Operations Research Society of China

Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry, etc. One of the main challenges usually is the non-convexity of the manifold constraints. By utilizing the geometry of manifold, a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold. From this perspective, intrinsic structures, optimality conditions and numerical algorithms for manifold optimization are investigated. Some recent progress on the theoretical results of manifold optimization is also presented.

Recently, alternating direction method of multipliers (ADMM) attracts much attentions from various fields and there are many variant versions tailored for different models. Moreover, its theoretical studies such as rate of convergence and extensions to nonconvex problems also achieve much progress. In this paper, we give a survey on some recent developments of ADMM and its variants.

We study the Fisher model of a competitive market from the algorithmic perspective. For that, the related convex optimization problem due to Gale and Eisenberg (Ann Math Stat 30(1):165–168, 1959) is used. The latter problem is known to yield a Fisher equilibrium under some structural assumptions on consumers’ utilities, e.g., homogeneity of degree 1, homotheticity. Our goal is to examine applicability of the convex optimization framework by departing from these traditional assumptions. We just assume the concavity of consumers’ utility functions. For this case, we suggest a novel concept of Fisher–Gale equilibrium by using consumers’ utility prices. The prices of utility transfer the utility of consumption bundle to a common numéraire. We develop a subgradient-type algorithm from Convex Analysis to compute a Fisher–Gale equilibrium via Gale’s approach. In order to decentralize prices, we additionally implement the auction design, i.e., consumers settle and update their individual prices and producers sell at the highest offer price. Our price adjustment is based on a tatonnement procedure, i.e., the prices change proportionally to consumers’ individual excess supplies. Historical averages of consumption are shown to clear the market of goods. Our algorithm is justified by a global rate of convergence. In the worst case, the number of price updates needed to achieve an $$\varepsilon $$ ε -tolerance is proportional to $$\frac{1}{\varepsilon ^2}$$ 1 ε 2 .