Stochastic mathematical programs with hybrid equilibrium constraints

AbstractThis paper considers a stochastic mathematical program with hybrid equilibrium constraints (SMPHEC), which includes either “here-and-now” or “wait-and-see” type complementarity constraints. An example is given to describe the necessity to study SMPHEC. In order to solve the problem, the sampling average approximation techniques are employed to approximate the expectations and smoothing and penalty techniques are used to deal with the complementarity constraints. Limiting behaviors of the proposed approach are discussed. Preliminary numerical experiments show that the proposed approach is applicable

Entropic approximation for mathematical programs with robust equilibrium constraints

In this paper, we consider a class of mathematical programs with robust equilibrium constraints represented by a system of semi-infinite complementarity constraints (SIC C). We propose a numerical scheme for tackling SICC. Specific ally, by relaxing the complementarity constraints and then randomizing the index set of SICC, we employ the well-known entropic risk measure to approximate the semi-infinite onstraints with a finite number of stochastic inequality constraints. Under some moderate conditions, we quantify the approximation in term s of the feasible set and the optimal value. The approximation scheme is then applied to a class of two stage stochastic mathematical programs with complementarity constraints in combination with the polynomial decision rules. Finally, we extend the discussion to a mathematical program with distributionally robust equilibrium constraints which is essentially a one stage stochastic program with semi-infinite stochastic constraints indexed by some probability measures from an ambiguity set defined through the KL-divergence

A Stackelberg Solution to Joint Optimization Problems: A Case Study of Green Design

AbstractDesign of complex engineered systems often involves optimization of multiple competing problems that are supposed to compromise to arrive at equilibrium optima, entailing a joint optimization problem. This paper reveals the leader-follower decision structure inherent in joint optimization problems. A Stackelberg game solution is formulated to model a leader-follower joint optimization problem as a two-level optimization problem between two decision makers, implicating a mathematical program that contains sub-optimization problems as its constraints. A case study of coffee grinder green design demonstrates the potential of Stackelberg solution to joint optimization of modularity subject with conflicting goals

A regularized variance-reduced modified extragradient method for stochastic hierarchical games

The theory of learning in games has so far focused mainly on games with simultaneous moves. Recently, researchers in machine learning have started investigating learning dynamics in games involving hierarchical decision-making. We consider an $N$-player hierarchical game in which the $i$th player's objective comprises of an expectation-valued term, parametrized by rival decisions, and a hierarchical term. Such a framework allows for capturing a broad range of stochastic hierarchical optimization problems, Stackelberg equilibrium problems, and leader-follower games. We develop an iteratively regularized and smoothed variance-reduced modified extragradient framework for learning hierarchical equilibria in a stochastic setting. We equip our analysis with rate statements, complexity guarantees, and almost-sure convergence claims. We then extend these statements to settings where the lower-level problem is solved inexactly and provide the corresponding rate and complexity statements

Modeling of Competition and Collaboration Networks under Uncertainty: Stochastic Programs with Resource and Bilevel

We analyze stochastic programming problems with recourse characterized by a bilevel structure. Part of the uncertainty in such problems is due to actions of other actors such that the considered decision maker needs to develop a model to estimate their response to his decisions. Often, the resulting model exhibits connecting constraints in the leaders (upper-level) subproblem. It is shown that this problem can be formulated as a new class of stochastic programming problems with equilibrium constraints (SMPEC). Sufficient optimality conditions are stated. A solution algorithm utilizing a stochastic quasi-gradient method is proposed, and its applicability extensively explained by practical numerical examples

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Validation of nominations in gas networks and properties of technical capacities

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