Online optimization with multiple budget constraints is challenging since the
online decisions over a short time horizon are coupled together by strict
inventory constraints. The existing manually-designed algorithms cannot achieve
satisfactory average performance for this setting because they often need a
large number of time steps for convergence and/or may violate the inventory
constraints. In this paper, we propose a new machine learning (ML) assisted
unrolling approach, called LAAU (Learning-Assisted Algorithm Unrolling), which
unrolls the online decision pipeline and leverages an ML model for updating the
Lagrangian multiplier online. For efficient training via backpropagation, we
derive gradients of the decision pipeline over time. We also provide the
average cost bounds for two cases when training data is available offline and
collected online, respectively. Finally, we present numerical results to
highlight that LAAU can outperform the existing baselines.

本文提出了一种新的机器学习辅助在线优化算法，并通过引入 Lagrangian 倍增方法，在线更新约束条件优化准则，实现在多个限制条件下进行深度学习模型联合优化，避免了现有算法的缺陷，提高了算法效率和性能。

基于学习的算法展开技术用于带预算约束的在线优化

Learning-Assisted Algorithm Unrolling for Online Optimization with Budget Constraints

This paper studies online optimization under inventory (budget) constraints.
While online optimization is a well-studied topic, versions with inventory
constraints have proven difficult. We consider a formulation of
inventory-constrained optimization that is a generalization of the classic
one-way trading problem and has a wide range of applications. We present a new
algorithmic framework, \textsf{CR-Pursuit}, and prove that it achieves the
minimal competitive ratio among all deterministic algorithms (up to a
problem-dependent constant factor) for inventory-constrained online
optimization. Our algorithm and its analysis not only simplify and unify the
state-of-the-art results for the standard one-way trading problem, but they
also establish novel bounds for generalizations including concave revenue
functions. For example, for one-way trading with price elasticity, the
\textsf{CR-Pursuit} algorithm achieves a competitive ratio that is within a
small additive constant (i.e., 1/3) to the lower bound of $\ln \theta+1$, where
$\theta$ is the ratio between the maximum and minimum base prices.

该研究提出了一种新的算法框架 CR-Pursuit，用于解决在线优化中的库存约束问题，并证明了在确定性算法中其取得最小的竞争比率。

库存约束下的竞争在线优化

Competitive Online Optimization under Inventory Constraints

In this paper we extend the market-making models with inventory constraints
of Avellaneda and Stoikov ("High-frequency trading in a limit-order book",
Quantitative Finance Vol.8 No.3 2008) and Gueant, Lehalle and Fernandez-Tapia
("Dealing with inventory risk", Preprint 2011) to the case of a rather general
class of mid-price processes, under either exponential or linear PNL utility
functions, and we add an inventory-risk-aversion parameter that penalises the
marker-maker if she finishes her day with a non-zero inventory. This general,
non-martingale framework allows a market-maker to make directional bets on
market trends whilst keeping under control her inventory risk. In order to
achieve this, the marker-maker places non-symmetric limit orders that favour
market orders to hit her bid (resp. ask) quotes if she expects that prices will
go up (resp. down).
With this inventory-risk-aversion parameter, the market-maker has not only
direct control on her inventory risk but she also has indirect control on the
moments of her PNL distribution. Therefore, this parameter can be seen as a
fine-tuning of the marker-maker's risk-reward profile.
In the case of a mean-reverting mid-price, we show numerically that the
inventory-risk-aversion parameter gives the market-maker enough room to tailor
her risk-reward profile, depending on her risk budgets in inventory and PNL
distribution (especially variance, skewness, kurtosis and VaR). For example,
when compared to the martingale benchmark, a market can choose to either
increase her average PNL by more than 15% and carry a huge risk, on inventory
and PNL, or either give up 5% of her benchmark PNL to increase her control on
inventory and PNL, as well as increasing her Sharpe ratio by a factor bigger
than 2.

在这篇论文中，我们将存货约束市场制作模型扩展到了相当一般的中间价格过程情况，针对指数或线性 PNL 效用函数，添加了一个存货风险规避参数，这个参数可以看作是市场制造者风险回报配置的微调。

具有库存约束和定向投注的高频市场制造

High-frequency market-making with inventory constraints and directional  bets

Market makers continuously set bid and ask quotes for the stocks they have
under consideration. Hence they face a complex optimization problem in which
their return, based on the bid-ask spread they quote and the frequency at which
they indeed provide liquidity, is challenged by the price risk they bear due to
their inventory. In this paper, we consider a stochastic control problem
similar to the one introduced by Ho and Stoll and formalized mathematically by
Avellaneda and Stoikov. The market is modeled using a reference price $S_t$
following a Brownian motion with standard deviation $\sigma$, arrival rates of
buy or sell liquidity-consuming orders depend on the distance to the reference
price $S_t$ and a market maker maximizes the expected utility of its P&L over a
finite time horizon. We show that the Hamilton-Jacobi-Bellman equations
associated to the stochastic optimal control problem can be transformed into a
system of linear ordinary differential equations and we solve the market making
problem under inventory constraints. We also shed light on the asymptotic
behavior of the optimal quotes and propose closed-form approximations based on
a spectral characterization of the optimal quotes.

本研究探讨了市场制造商在股票交易中所面临的复杂优化问题，通过建立随机控制问题的模型，研究了制造商的收益与风险的平衡，得到了一组线性常微分方程的解，并基于对最优行情的谱特性提出了一种简单的近似方法。