# Alejandro JofreUniversity of Chile · Centro de Modelamiento Matemático (CMM)

Alejandro Jofre

PhD Applied Mathematics France

## About

73

Publications

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1,443

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Citations since 2017

Introduction

Additional affiliations

January 1990 - present

## Publications

Publications (73)

We discuss a procurement problem with transportation losses and piecewise linear production costs.We first provide an algorithm based on Knaster-Tarski's fixed point theorem to solve the allocationproblem in the quadratic losses case. We then identify a monotony condition on the types distribution under which the Bayesian cost minimizing mechanism...

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 mechanis...

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 co...

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 cu...

For underground mine, the current usual technique for ore extraction is block caving, which generates and induces seismic activity in the mine. To understand block caving method is one of the most challenging problems in underground mining. This method relies on gravity to break and transport large amounts of ore and waste. The state of art in dama...

Block caving is an ore extraction technique used in the copper mines of Chile. It uses gravity to ease the breaking of rocks, and to facilitate the extraction from the mine of the resulting mixture of ore and waste. To simulate this extraction process numerically and better understand its impact on the mine environment, we study 3 variational model...

Worldwide the mining activity is facing lower grades, deeper ore bodies and stronger stresses within increasing massive operations. These challenges have several technical difficulties, uncertainties and associated systemic risks, which can affect the business survival and the long-term sustainability of the operation. For underground mine, the cur...

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....

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 lopside...

We propose stochastic approximation (SA) methods for stochastic smooth convex optimization (SSCO) with a general distribution. Typically, it is assumed an oracle with an upper bound $\sigma^2$ on its variance (OUBV). Differently, we suppose an oracle with \emph{multiplicative noise} (OMN), a setting rarely addressed before. This is a more aggressiv...

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...

We consider stochastic variational inequalities with monotone operators defined as the expected value of a random operator. We assume the feasible set is the intersection of a large family of convex sets. We propose a method that combines stochastic approximation with incremental constraint projections meaning that at each iteration, a step similar...

We propose stochastic extragradient methods for stochastic variational inequalities with a linear search requiring only pseudo-monotonicity of the operator and no knowledge of the Lipschitz constant $L$. We provide convergence and complexity analysis, allowing for an unbounded feasible set, unbounded operator, non-uniform variance of the oracle and...

A theory of economic equilibrium for incomplete financial markets in general real assets is developed in a new formulation with currency-denominated prices. The “goods” are not only commodities, and they can influence utility through retention as an alternative to consumption. Perfect foresight is relinquished in a rolling horizon approach to marke...

In this chapter, 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 stemming from these models are the computation of an optimal allocation for the Independent System Operator, the study of equilibria (existence and uniqueness in parti...

Credit scoring is an automated, objective and consistent tool which helps lenders to provide quick loan decisions. It can replace some of the more mechanical work done by experienced loan officers whose decisions are intuitive but potentially subject to bias. Prospective borrowers may have a strong motivation to fraudulently falsify one or more of...

The share of the services offered via the Internet by nowadays banking companies is quickly growing, making of the understanding of online customers one of the major concerns. Data mining tools have proven their efficiency in addressing this challenge by providing unsupervised quantitative techniques to identify those segments of customers with sim...

Forestry has contributed many problems to the Operations Research (OR) community. At the same time, OR has developed many models and solution methods for use in forestry. In this article, we describe the current status of research on the application of OR methods to forestry and a number of research challenges or open questions that we believe will...

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...

Convexity has long had an important role in economic theory, but some recent developments have featured it all the more in problems of equilibrium. Here the tools of convex analysis are applied to a basic model of incomplete financial markets in which assets are traded and money can be lent or borrowed between the present and future. The existence...

Convexity has long had an important role in economic theory, but some recent developments have featured it all the more in problems of equilibrium. Here the tools of convex analysis are applied to a basic model of financial markets in which assets are traded and money can be lent or borrowed between the present and future. The existence of an equil...

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 ac...

It is shown that a number of variational and equilibrium problems can be cast as finding the maxinf-points or minsup-points of bivariate functions, for short, bifunctions. These problems include linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, noncooperative games, and Walras and Nash equilibrium pr...

The Working Group had been given the mandate of mainly looking into questions from Sects. 2 and 6 of the Discussion Document, i.e., questions like: How can mathematics, especially industrial mathematics, be made more visible to the public at large? How can mathematics be made more appealing and exciting to students and the professionals in industry...

In an economic model of exchange of goods, the preference structure can be specified by utility functions. Under utility conditions identified here more broadly than usual, except for concavity in place of quasi-concavity, every equilibrium will be stable in a doubly local sense with respect to shifts in the agent's holdings and Walrasian tâtonneme...

The behavior of DC load flow formulations when they are used in economic
dispatch and nodal pricing models is discussed. It is demonstrated that nonnegative
prices in these models are sufficient to guarantee global optimality of any
local optimum, even if the feasible region is not convex, and so a negative nodal price
is an indicator of a possible...

In the models of multi-stage equilibrium with uncertain financial markets that have so far been formulated in extension of
the classical Walrasian model with only a single stage, each state is completely isolated in its activity. If there is production,
it ends in the state in which it begins. Goods that are not consumed within a state merely peris...

Abstract—The behaviour,of DC Load-�ow formulations,when they are used in economic,dispatch,and nodal pricing models,is discussed. It is demonstrated,that non-negative prices in these models,are suf�cient to guarantee,global optimality of any local optimum, even if the feasible region is not convex, and so a negative nodal price is an indicator of a...

In this paper we prove a general version of the Second Welfare Theorem for a non-convex and non-transitive economy, with public goods and other externalities in consumption. For this purpose we use the sub-gradient to the distance function (normal cone) to define the pricing rule in this general context.

We explore convergence notions for bivariate functions that yield convergence and stability results for their maxinf (or minsup) points. This lays the foundations for the study of the stability of solutions to variational inequalities, the solutions of inclusions, of Nash equilibrium points of non-cooperative games and Walras economic equilibrium p...

We consider a pool type electricity market in which generators bid prices in a sealed bid form and are dispatched by an independent
system operator (ISO). In our model, demand is inelastic and the ISO allocates production to minimize the system costs while
considering the transmission constraints. In a departure from received literature, the model...

This paper models an electricity market with generators and customers located on a network. Generators bid cost functions and are dispatched by a system operator that minimizes the system cost while considering network constraints. We prove the existence of equilibrium and show that transmission constraints render the market imperfectly competitive...

Publicación ISI Email : ajofre@dim.uchile.cl; rtr@math.washington.edu; rjbwets@ucdavis.edu Variational inequality representations are set up for a general Walrasian model of consumption and production with trading in a market. The variational inequalities are of functional rather than geometric type and therefore are able to accommodate a wider ran...

We extend the traditional two-stage linear stochastic program by probabilistic constraints imposed in the second stage. This adds nonlinearity such that basic arguments for analyzing the structure of linear two-stage stochastic programs have to be rethought from the very beginning. We identify assumptions under which the problem is structurally sou...

In this work by using nonsmooth analysis techniques we provide a geometric characterization of both the free disposal hypothesis for production sets and the strict monotonicity condition for preference relations even for nonconvex economies.

In this paper we proved a nonconvex separation property for general sets which coincides with the Hahn-Banach separation theorem when sets are convexes. Properties derived from the main result are used to compute the subgradient set to the distance function in special cases and they are also applied to extending the Second Welfare Theorem in econom...

In this paper we model and analyze a market equilibrium structure working on a network. The model is motivated by competition
in electricity power generation markets, where consumers and producers are located in different nodes connected by power transmission
lines. We analyze two different equilibrium concepts, namely, the Walrasian and the noncoo...

In this paper, we prove a new version of the Second Welfare Theorem for economies with a finite number of agents and an infinite number of commodities, when the preference correspondences are not convex-valued and/or when the total production set is not convex. For this kind of nonconvex economies, a recent result, obtained by one of the authors, i...

A convex, compact, and possibly discontinuous better reply secure game has a Nash equilibrium. We introduce a very weak notion of continuity that can be used to establish that a game is better reply secure and we show that this notion of continuity is satisfied by a large class of games. Copyright The Econometric Society 2006.

This work presents a novel day-ahead energy acquisition model for a distribution company (DisCo) in a competitive market based on Pool and financial bilateral contracts. The market structure encompasses wholesale generation companies, distributed generation (DG) units of independent producers, DG units owned by the DisCo, and load curtailment optio...

In this paper we model and analyze a market equilibrium structure working on a network. The model is motivated by competition in electricity power generation markets, where con-sumers and producers are located in different nodes connected by power transmission lines. We analyze two different equilibrium concepts, namely, the Walrasian and the nonco...

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 limi...

We study some Bolza-type problems governed by two classes of functional evolution inclusions where the controls are Young measures. In particular, we present some variational properties of the value function associated to these control problems, and we show that the lower value function is a viscosity subsolution of the associated Hamilton-Jacobi-B...

We explore convergence notions for bivariate functions that yield convergence and stability results for their max/inf points. The results are then applied to obtain continuity results for Walras equilibrium points under perturbations of the utility functions of the agents.

The purpose of this paper is to study the continuity and uniqueness properties of equilibria for linear exchange economies. We characterize the sets of utility vectors and initial endowments for which the equilibrium price is unique and respectively the set for which the equilibrium allocation is unique. We show that the equilibrium allocation corr...

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 analyti...

In this paper a two-level model and optimization algorithms areintroduced to assist forestry companies in simultaneously considering strategicinvestment and tactical planning decisions. A procedure to reduce thediscrepancy produced in the aggregation and disaggregation process used to linkthese two-level decisions is also presented. This procedure...

We establish, in infinite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension
of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers
and show the relation of our property with the extremal principle of Mordukhovich.

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 infinitel...

We establish, in infinite dimensional Banach space, a nonconvex separation property for general closed sets that is an extension of Hahn-Banach separation theorem. We provide some consequences in optimization, in particular the existence of singular multipliers and show the relation of our principle with the extremal principle of Mordukhovich.

We show how our method in [Numer. Funct. Anal. Optimization 15, No. 5-6, 531-535 (1994; Zbl 0807.49015)] can be applied in a simple way to characterize convex functions in terms of the monotonicity of any pre-subdifferential.

We prove that a lower semicontinuous function defined on a Banach space is convex if and only if its subdifferential ismonotone.

We prove that a lower semicontinuous function defined on a reflexive Banach space is convex if and only if its Clarke subdifferential is monotone.

Subdifferentials of convex functions and some regular functions f are expressed in terms of limiting gradients at points in a given dense subset of $\operatorname{dom} \nabla f$.

Subdifferentials of convex functions and some regular functions f are expressed in terms of limiting gradients at points in a given dense subset of dorn ∇f.

New notions of tangent cones which have recently been introduced are compared. These notions are variants of Clarke’s strict tangent cone and give rise to corresponding generalized derivatives. They are closed, convex and larger than the Clarke strict tangent cone, and these are desirable features.

In this paper, we introduce a new class of nonsmooth functions in terms of a continuity property of the usual directional derivative. Under this approach, we study the subregular and the semismooth functions. Finally, we give conditions for a marginal function to be subregular and semismooth.

Given a real valued function f, defined on a locally convex topological space X, locally Lipschitzian, and Gateaux-differentiable on a dense subset D in X, we have studied under what hypotheses Charke's generalized gradient can be written as
¶f(x) = [`(co)] \text { w* limy ® x Ñf(y)/ y Î D} \text ,\partial f(x) = \overline {co} {\text{ }}\{ w^* \...

This paper considers an electricity spot market consisting of a network, a set of producers located in nodes of the network, and a central agent. Production is organized by means of an auction. Once firms simultaneously bid cost functions, the central agent decides the quantity each firm produces and the flows through the network lines. The purpose...

In the prevailing theory of economic equilibrium with incomplete markets, assets pay in "units of account" which are regarded as money but have no link to the actual currencies that rule in financial dealings. The units of account at any given time are unrelated to those at another time or in another state, as if the money in question must be dispo...

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 represen- tation. Epi-convergence arguments are employed to show that, in the li...

Abstract An SLA/Contract is an agreement between a client and a service provider. It species

In classical Walrasian equilibrium, everything is compressed into a single time period: all production, trade, and consumption. Eorts at expanding to a framework with uncertain future states beyond a present state have generally mimicked this pattern by insisting that, apart from possibilities opened by financial contracts, all the activity in each...

We prove an extension of the second welfare theorem for nonconvex economies with an infinite number of goods, that is, when the preference set-valued maps or the production sets can be nonconvex. Firstly, it is proved a “viscous” version of this result without any compactness or epi-lipschitzianity assumption, secondly it is deduced the exact versi...

## Projects

Projects (2)

We wish to study optimization and variational inequality problems when (1) objective functions, operators and constraints are described by expectations with respect to a probability distribution; (2) these expectations cannot be computed directly; but (3) one can obtain a random i.i.d. sample of size N (or an infinite sample) from that distribution. Main goals include achieving optimal or nearly optimal complexity in terms of sample size and/or other resources; including constraints that are also given by expectations; and dealing with heavier tailed data than what the typical sub-Gaussian assumptions in the literature.