Papers by Alejandro Jofré
Procurement Auctions with Losses
To study first-price procurement auctions in the presence of losses, we introduce a new fixed poi... more To study first-price procurement auctions in the presence of losses, we introduce a new fixed point gradient flow algorithm to compute the Bayesian Nash Equilibrium. We use this efficient algorithm to compare optimal, first-price and VCG auctions. This allows us to numerically estimate the social cost of sub-optimality of the nodal pricing mechanism in wholesale electricity markets. We also derive a closed form expression of the optimal mechanism procurement cost when the types are uniformly distributed. Last we show how the algorithm adapts to more general grids.
Cornell University - arXiv, Mar 1, 2017
We propose dynamic sampled stochastic approximated (DS-SA) extragradient methods for stochastic v... more We propose dynamic sampled stochastic approximated (DS-SA) extragradient methods for stochastic variational inequalities (SVI) that are robust with respect to an unknown Lipschitz constant L. We propose, to the best of our knowledge, the first provably convergent robust SA method with variance reduction, either for SVIs or stochastic optimization, assuming just an unbiased stochastic oracle and a large sample regime. This widens the applicability and improves, up to constants, the desired efficient acceleration of previous variance reduction methods, all of which still assume knowledge of L (and, hence, are not robust against its estimate). Precisely, compared to the iteration and oracle complexities of O(ǫ −2) of previous robust methods with a small stepsize policy, our robust method uses a DS-SA line search scheme obtaining the faster iteration complexity of O(ǫ −1) with oracle complexity of (ln L)O(dǫ −2) (up to log factors on ǫ −1) for a d-dimensional space. This matches, up to constants, the sample complexity of the sample average approximation estimator which does not assume additional problem information (such as L). Differently from previous robust methods for ill-conditioned problems, we allow an unbounded feasible set and an oracle with multiplicative noise (MN) whose variance is not necessarily uniformly bounded. These properties are appreciated in our complexity estimates which depend only on L and local variances or forth moments at solutions x *. The robustness and variance reduction properties of our DS-SA line search scheme come at the expense of nonmartingale-like dependencies (NMD) due to the needed inner statistical estimation of a lower bound for L. In order to handle a NMD and a MN, our proofs rely on a novel localization argument based on empirical process theory. Additionally, we propose a second provable convergent method for SVIs over the wider class of Hölder continuous operators without any knowledge of its endogenous parameters.
Cornell University - arXiv, Jul 23, 2019
Motivated by the problem of market power in electricity markets, we introduced in previous works ... more Motivated by the problem of market power in electricity markets, we introduced in previous works a mechanism for simplified markets of two agents with linear cost. In standard procurement auctions, the market power resulting from the quadratic transmission losses allows the producers to bid above their true values, which are their production cost. The mechanism proposed in the previous paper optimally reduces the producers' margin to the society's benefit. In this paper, we extend those results to a more general market made of a finite number of agents with piecewise linear cost functions, which makes the problem more difficult, but simultaneously more realistic. We show that the methodology works for a large class of externalities. We also provide an algorithm to solve the principal allocation problem. Our contribution provides a benchmark to assess the sub-optimality of the mechanisms used in practice.
Cornell University - arXiv, Jul 16, 2021
Understanding the strategic behavior of miners in a blockchain is of great importance for its pro... more Understanding the strategic behavior of miners in a blockchain is of great importance for its proper operation. A common model for mining games considers an infinite time horizon, with players optimizing asymptotic average objectives. Implicitly, this assumes that the asymptotic behaviors are realized at human-scale times, otherwise invalidating current models. We study the mining game utilizing Markov Decision Processes. Our approach allows us to describe the asymptotic behavior of the game in terms of the stationary distribution of the induced Markov chain. We focus on a model with two players under immediate release, assuming two different objectives: the (asymptotic) average reward per turn and the (asymptotic) percentage of obtained blocks. Using tools from Markov chain analysis, we show the existence of a strategy achieving slow mixing times, exponential in the policy parameters. This result emphasizes the imperative need to understand convergence rates in mining games, validating the standard models. Towards this end, we provide upper bounds for the mixing time of certain meaningful classes of strategies. This result yields criteria for establishing that long-term averaged functions are coherent as payoff functions. Moreover, by studying hitting times, we provide a criterion to validate the common simplification of considering finite states models. For both considered objectives functions, we provide explicit formulae depending on the stationary distribution of the underlying Markov chain. In particular, this shows that both mentioned objectives are not equivalent. Finally, we perform a market share case study in a particular regime of the game. More precisely, we show that an strategic player with a sufficiently large processing power can impose negative revenue on honest players.
Cornell University - arXiv, Nov 26, 2021
In this paper we study a pollution regulation problem in an electricity market with a network str... more In this paper we study a pollution regulation problem in an electricity market with a network structure. The market is ruled by an independent system operator (ISO for short) who has the goal of reducing the pollutant emissions of the providers in the network, by encouraging the use of cleaner technologies. The problem of the ISO formulates as a contracting problem with each one of the providers, who interact among themselves by playing a stochastic differential game. The actions of the providers are not observable by the ISO which faces moral hazard. By using the dynamic programming approach, we represent the value function of the ISO as the unique viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation. We prove that this solution is smooth and characterise the optimal controls for the ISO. Numerical solutions to the problem are presented and discussed. We consider also a simpler problem for the ISO, with constant production levels, that can be solved explicitly in a particular setting.
Research Papers in Economics, Jul 1, 2019
Motivated by the problem of market power in electricity markets, we introduced in previous works ... more Motivated by the problem of market power in electricity markets, we introduced in previous works a mechanism for simplified markets of two agents with linear cost. In standard procurement auctions, the market power resulting from the quadratic transmission losses allows the producers to bid above their true values, which are their production cost. The mechanism proposed in the previous paper optimally reduces the producers' margin to the society's benefit. In this paper, we extend those results to a more general market made of a finite number of agents with piecewise linear cost functions, which makes the problem more difficult, but simultaneously more realistic. We show that the methodology works for a large class of externalities. We also provide an algorithm to solve the principal allocation problem. Our contribution provides a benchmark to assess the sub-optimality of the mechanisms used in practice.
Variational Analysis and Applications
The existence of an equilibrium in an extended Walrasian economic model of exchange is confirmed ... more The existence of an equilibrium in an extended Walrasian economic model of exchange is confirmed constructively by an iterative scheme. In this scheme, truncated variational inequality problems are solved in which the agents' budget constraints are relaxed by a penalty representation. Epi-convergence arguments are employed to show that, in the limit, a virtual equilibrium is obtained, if not actually a classical equilibrium. A number of technical hurdles are, in this way, surmounted.
Strengthening Mathematics in the Developing World
Proceedings of the International Congress of Mathematicians (ICM 2018), 2019
Bilevel optimization applied to strategic pricing in electricity markets and extension to markets with massive entry of renewable energies and distributed generation
International audienc
We present some key aspects of wholesale electricity markets modeling and more specifically focus... more We present some key aspects of wholesale electricity markets modeling and more specifically focus our attention on auctions and mechanism design. Some of the results arising from those models are the computation of an optimal allocation for the Independent System Operator, the study of the equilibria (existence and unicity in particular) and the design of mechanisms to increase the social surplus. From a more general perspective, this field of research provides clues to discuss how wholesale electricity market should be regulated. We start with a general introduction and then present some results the authors obtained recently. We also briefly expose some undergoing related work. As an illustrative example, a section is devoted to the computation of the Independent System Operator response function for a symmetric binodal setting with piece-wise linear production cost functions.
Journal of Optimization Theory and Applications, 2001
The purpose of this paper is to study the differentiability properties of equilibrium prices and ... more The purpose of this paper is to study the differentiability properties of equilibrium prices and allocations in a linear exchange economy when the initial endowments and utility vectors vary. We characterize an open dense subset of full measure of the initial endowment and utility vector space on which the equilibrium price vector is a real analytic function, hence infinitely differentiable function. We provide an explicit formula to compute the equilibrium price and allocation around a point where it is known. We also show that the equilibrium price is a locally Lipschitzian mapping on the whole space. Finally, using the notion of the Clarke generalized gradient, we prove that linear exchange economies satisfy a property of gross substitution.
Mathematical Programming, 2018
We propose dynamic sampled stochastic approximation (SA) methods for stochastic optimization with... more We propose dynamic sampled stochastic approximation (SA) methods for stochastic optimization with a heavy-tailed distribution (with finite 2nd moment). The objective is the sum of a smooth convex function with a convex regularizer. Typically, it is assumed an oracle with an upper bound σ 2 on its variance (OUBV). Differently, we assume an oracle with multiplicative noise. This rarely addressed setup is more aggressive but realistic, where the variance may not be bounded. Our methods achieve optimal iteration complexity and (near) optimal oracle complexity. For the smooth convex class, we use an accelerated SA method a la FISTA which achieves, given tolerance ǫ > 0, the optimal iteration complexity of O(ǫ − 1 2) with a near-optimal oracle complexity of O(ǫ −2)[ln(ǫ − 1 2)] 2. This improves upon Ghadimi and Lan [Math. Program., 156:59-99, 2016] where it is assumed an OUBV. For the strongly convex class, our method achieves optimal iteration complexity of O(ln(ǫ −1)) and optimal oracle complexity of O(ǫ −1). This improves upon Byrd et al. [Math. Program., 134:127-155, 2012] where it is assumed an OUBV. In terms of variance, our bounds are local: they depend on variances σ(x *) 2 at solutions x * and the per unit distance multiplicative variance σ 2 L. For the smooth convex class, there exist policies such that our bounds resemble those obtained if it was assumed an OUBV with σ 2 := σ(x *) 2. For the strongly convex class such property is obtained exactly if the condition number is estimated or in the limit for better conditioned problems or for larger initial batch sizes. In any case, if it is assumed an OUBV, our bounds are thus much sharper since typically max{σ(x *) 2 , σ 2 L } ≪ σ 2. 1 Typical reasons are: a sample space with high dimension requiring Monte Carlo evaluation, no knowledge of the distribution P or, even worse, no knowledge of a close form for F .
Variance-Based Extragradient Methods with Line Search for Stochastic Variational Inequalities
SIAM Journal on Optimization, 2019
In this paper, we propose dynamic sampled stochastic approximated (DS-SA) extragradient methods f... more In this paper, we propose dynamic sampled stochastic approximated (DS-SA) extragradient methods for stochastic variational inequalities (SVIs) that are robust with respect to an unknown Lipschitz c...
SIAM Journal on Optimization, 2017
We propose an extragradient method with stepsizes bounded away from zero for stochastic variation... more We propose an extragradient method with stepsizes bounded away from zero for stochastic variational inequalities requiring only pseudo-monotonicity. We provide convergence and complexity analysis, allowing for an unbounded feasible set, unbounded operator, non-uniform variance of the oracle and, also, we do not require any regularization. Alongside the stochastic approximation procedure, we iteratively reduce the variance of the stochastic error. Our method attains the optimal oracle complexity O(1/ǫ 2) (up to a logarithmic term) and a faster rate O(1/K) in terms of the mean (quadratic) natural residual and the D-gap function, where K is the number of iterations required for a given tolerance ǫ > 0. Such convergence rate represents an acceleration with respect to the stochastic error. The generated sequence also enjoys a new feature: the sequence is bounded in L p if the stochastic error has finite p-moment. Explicit estimates for the convergence rate, the oracle complexity and the p-moments are given depending on problem parameters and distance of the initial iterate to the solution set. Moreover, sharper constants are possible if the variance is uniform over the solution set or the feasible set. Our results provide new classes of stochastic variational inequalities for which a convergence rate of O(1/K) holds in terms of the mean-squared distance to the solution set. Our analysis includes the distributed solution of pseudo-monotone Cartesian variational inequalities under partial coordination of parameters between users of a network.
Computational Economics, 2017
We described a method to solve deterministic and stochastic Walras equilibrium models based on as... more We described a method to solve deterministic and stochastic Walras equilibrium models based on associating with the given problem a bifunction whose maxinf-points turn out to be equilibrium points. The numerical procedure relies on an augmentation of this bifunction. Convergence of the proposed procedure is proved by relying on the relevant lopsided convergence. In the dynamic versions of our models, deterministic and stochastic, we are mostly concerned with models that equip the agents with a mechanism to transfer goods from one time period to the next, possibly simply savings, but also allows for the transformation of goods via production.
Continuity and Differentiability of Equilibria for Linear Exchange Economies II
The purpose of this paper is to study how the equilibrium prices and allocations in a linear exch... more The purpose of this paper is to study how the equilibrium prices and allocations in a linear exchange economy vary with respect to the intial endowments and utility vectors. We characterize an open dense subject of full measure of the initial endowment and utility vector space on which the equilibrium price vector is a real-analytic hence infinitely differentiable function. We
SIAM Journal on Optimization, 2015
Our aim in this paper is to prove geometric characterizations of the free disposal condition for ... more Our aim in this paper is to prove geometric characterizations of the free disposal condition for nonconvex economies on infinite dimensional commodity spaces even if the cone and the production set involved in the condition have an empty interior such as in L 1 with the positive cone L 1 +. We then use this characterization to prove the existence of Pareto and weak Pareto optimal points. We also explore a notion of extremal systemsà la Kruger-Mordukhovich. We show that the free disposal hypothesis alone assures the extremality of the production set with respect to some set.
SSRN Electronic Journal, 2014
A theory of general economic equilibrium with incomplete financial markets is developed with many... more A theory of general economic equilibrium with incomplete financial markets is developed with many new features, including currency-denominated prices which enable treatment of currency-based derivative instruments and collateralized contracts. Prices in such models with standard market structure have previously been articulated only in "units of account" which have no link to an actual currency and are subject to indeterminancy in scaling. That shortcoming, which prevents ordinary price comparisons between different states, present and future, has stemmed from a focus on consumption as the sole source of economic value, but here retention of goods is allowed to influence their utility as well. The "goods" are not just commodities and thus can encompass other elements essential to finance. The framework is that of an economy operating in a currency agents find attractive to retain, in balance with other needs. The attractiveness comes from Keynesian considerations about uncertainty which until now have not been brought in. An altered view of time and states helps by loosening the grip of perfect foresight in future markets. Existence is established with a single currency denominating the units of account in all states, and price indeterminancy is thereby removed. All contracts issued in the financial markets can be interpreted then as "real contracts." Endogenously generated transaction costs on sales of contracts keep the financial markets from getting out of hand and lead to bid-ask spreads, including a gap between interest rates for lending and borrowing money. To this end, equilibrium is given a variational formulation that brings fresh tools to the subject. A different way of proving existence in that setting, not merely in a generic sense and without normalizing to a price simplex or arbitrarily fixing "price levels" in the future states, makes use of duality bounds for the budget constraints. In the currency framework of the model, the proof of equilibrium is able moreover to proceed under far weaker assumptions than usual on the agents' preferences and endowments.
Mathematical Programming, 2014
We consider the solution of strongly monotone variational inequalities of the form F (x *) (x − x... more We consider the solution of strongly monotone variational inequalities of the form F (x *) (x − x *) ≥ 0, for all x ∈ X. We focus on special structures that lend themselves to sampling, such as when X is the intersection of a large number of sets, and/or F is an expected value or is the sum of a large number of component functions. We propose new methods that combine elements of incremental constraint projection and stochastic gradient. We analyze the convergence and the rate of convergence of these methods with various types of sampling schemes, and we establish a substantial rate of convergence advantage for random sampling over cyclic sampling.
SIAM Journal on Optimization, 2014
It's shown that a number of variational problems can be cast as finding the maxinfpoints (or mins... more It's shown that a number of variational problems can be cast as finding the maxinfpoints (or minsup-points) of bifunctions (= bivariate functions). These variational problems include: linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, noncooperative games, Walras and Nash equilibrium problems. One can then appeal to the theory of lopsided convergence for bifunctions to derive stability results for each one of these variational problems.
Uploads
Papers by Alejandro Jofré