Desmond W. H. Cai

A Unified Analysis of Max-Min Weighted SINR for MIMO Downlink System

This paper studies the max-min weighted signal-to-interference-plus-noise ratio (SINR) problem in the multiple-input-multiple-output (MIMO) downlink, where multiple users are weighted according to priority and are subject to a weighted-sum-power constraint. First, we study the multiple-input-single-output (MISO) and single-input-multiple- output (SIMO) problems using nonlinear Perron-Frobenius theory. As a by-product, we solve the open problem of convergence for a previously proposed MISO algorithm by Wiesel, Eldar, and Shamai in 2006. Furthermore, we unify our analysis with respect to the previous alternate optimization algorithm proposed by Tan, Chiang, and Srikant in 2009, by showing that our MISO result can, in fact, be derived from their algorithm. Next, we combine our MISO and SIMO results into an algorithm for the MIMO problem. We show that our proposed algorithm is optimal when the channels are rank-one, or when the network is operating in the low signal-to-noise ratio (SNR) region. Finally, we prove the parametric continuity of the MIMO problem in the power constraint, and we use this insight to propose a heuristic initialization strategy for improving the performance of our (generally) suboptimal MIMO algorithm. The proposed initialization strategy exhibits improved performance over random initialization.

Max-Min SINR Coordinated Multipoint Downlink Transmission—Duality and Algorithms

This paper considers the max-min weighted signal-to-interference-plus-noise ratio (SINR) problem subject to multiple weighted-sum power constraints, where the weights can represent relative power costs of serving different users. First, we study the power control problem. We apply nonlinear Perron-Frobenius theory to derive closed-form expressions for the optimal value and solution and an iterative algorithm which converges geometrically fast to the optimal solution. Then, we use the structure of the closed-form solution to show that the problem can be decoupled into subproblems each involving only one power constraint. Next, we study the multiple-input-single-output (MISO) transmit beamforming and power control problem. We use uplink-downlink duality to show that this problem can be decoupled into subproblems each involving only one power constraint. We apply this decoupling result to derive an iterative subgradient projection algorithm for the problem.

The role of a market maker in networked cournot competition

We study the role of a market maker (or market operator) in a transmission constrained electricity market. We model the market as a one-shot networked Cournot competition where generators supply quantity bids and load serving entities provide downward sloping inverse demand functions. This mimics the operation of a spot market in a deregulated market structure. In this paper, we focus on possible mechanisms employed by the market maker to balance demand and supply. In particular, we consider three candidate objective functions that the market maker optimizes - social welfare, residual social welfare, and consumer surplus. We characterize the existence of Generalized Nash Equilibrium (GNE) in this setting and demonstrate that market outcomes at equilibrium can be very different under the candidate objective functions.

Optimal max-min fairness rate control in wireless networks: Perron-Frobenius characterization and algorithms

Rate adaptation and power control are two key resource allocation mechanisms in multiuser wireless networks. In the presence of interference, how do we jointly optimize end-to-end source rates and link powers to achieve weighted max-min rate fairness for all sources in the network? This optimization problem is hard to solve as physical layer link rate functions are nonlinear, nonconvex, and coupled in the transmit powers. We show that the weighted max-min rate fairness problem can, in fact, be decoupled into separate fairness problems for flow rate and power control. For a large class of physical layer link rate functions, we characterize the optimal solution analytically by a nonlinear Perron-Frobenius theory (through solving a conditional eigenvalue problem) that captures the interaction of multiuser interference. We give an iterative algorithm to compute the optimal flow rate that converges geometrically fast without any parameter configuration. Numerical results show that our iterative algorithm is computationally fast for both the Shannon capacity, CDMA, and piecewise linear link rate functions.

Coordinated Max-Min SIR Optimization in Multicell Downlink - Duality and Algorithm

Typical formulations of max-min weighted SIR problems involve either a total power constraint or individual power constraints. These formulations are unable to handle the complexities in multicell networks where each base station can be subject to its own sum power constraint. This paper considers the max-min weighted SIR problem subject to multiple weighted-sum power constraints, where the weights can represent relative power costs of serving different users. First, we derive the uplink-downlink duality principle by applying Lagrange duality to the single-constraint problem. Next, we apply nonlinear Perron-Frobenius theory to derive a closed-form solution for the multiple-constraint problem. Then, by exploiting the structure of the closed-form solution, we relate the multiple-constraint problem with its single-constraint subproblems and establish the dual uplink problem. Finally, we further apply nonlinear Perron-Frobenius theory to derive an algorithm which converges geometrically fast to the optimal solution.

Resistive Network Optimal Power Flow: Uniqueness and Algorithms

The optimal power flow (OPF) problem minimizes the power loss in an electrical network by optimizing the voltage and power delivered at the network buses, and is a nonconvex problem that is generally hard to solve. By leveraging a recent development on the zero duality gap of OPF, we propose a second-order cone programming convex relaxation of the resistive network OPF, and study the uniqueness of the optimal solution using differential topology, especially the Poincare-Hopf Index Theorem. We characterize the global uniqueness for different network topologies, e.g., line, radial, and mesh networks. This serves as a starting point to design distributed local algorithms with global behaviors that have low complexity, are computationally fast, and can run under synchronous and asynchronous settings in practical power grids.

Optimal contract for wind power in day-ahead electricity markets

The growth of wind energy production poses several challenges in its integration in current electric power systems. In this work, we study how a wind power producer can bid optimally in existing electricity markets. We derive optimal contract size and expected profit for a wind producer under arbitrary penalty function and generation costs. A key feature of our analysis is to allow for the wind producer to strategically withhold production once the day ahead contract is signed. Such strategic behavior is detrimental to the smooth functioning of electricity markets. We show that under simple conditions on the offered price and marginal imbalance penalty, a risk neutral profit maximizing wind power producer will produce as much as wind power is available (up to its contract size).

Max-min weighted SINR in coordinated multicell MIMO downlink

This paper studies the optimization of a multicell multiple-input-single-output (MISO) downlink system in which each base station serves multiple users, and each user is served by only one base station. First, we consider the problem of maximizing the minimum weighted signal-to-interference-plus-noise ratio (SINR) of all users subject to a single weighted-sum power constraint, where the weights can represent relative power costs of serving different users in each cell. We apply concave Perron-Frobenius theory to propose a joint power control and linear beamforming algorithm which converges geometrically fast to the optimal solution. As a by-product, we resolve an open problem of convergence of a previously proposed algorithm by Wiesel, Eldar, and Shamai in 2006. Next, we study the max-min weighted SINR problem subject to multiple weighted-sum power constraints and we show that it can be decoupled into its associated single-constrained subproblems.

Inefficiency in forward markets with supply friction

The growth of renewable resources will introduce significant variability and uncertainty into the grid. It is likely that “peaker” plants will be a crucial dispatchable resource for compensating for the variations in renewable supply. Thus, it is important to understand the strategic incentives of peaker plants and their potential for exploiting market power due to having responsive supply. To this end, we study an oligopolistic two-settlement market comprising of two types of generation (baseloads and peakers) where there is perfect foresight. We characterize symmetric equilibria in this context via closed-form expressions. However, we also show that, when the system is capacity-constrained, there may not exist equilibria in which baseloads and peakers play symmetric strategies. This happens because of opportunities for both types of generation to exploit market power to increase prices.

Optimal investment of conventional and renewable generation assets

Driven by the national policy to expand renewable generation, as well as the advances in renewable technologies that reduce the cost of small-scale renewable generation units, distributed generation at end users will comprise a significant fraction of electricity generation in the future. We study the problem faced by a social planner who seeks to minimize the long-term discounted costs (associated with both the procurement and the usage of conventional and distributed generation assets), subject to meeting an inelastic demand for electricity. Under mild conditions on the problem parameters, we fully characterize the optimal investment policy for the social planner. We also analyze the impact of problem parameters (e.g., asset lifespans) on the optimal investment policy through numerical examples.