Directional Necessary Optimality Conditions for Bilevel Programs

The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian–Fromovitz constraint qualification (MFCQ). In this paper, we show that even the first order sufficient condition for metric subregularity (which is, in general, weaker than MFCQ) fails at each feasible point of the bilevel program. We introduce the concept of a directional calmness condition and show that, under the directional calmness condition, the directional necessary optimality condition holds. Although the directional optimality condition is, in general, sharper than the nondirectional one, the directional calmness condition is, in general, weaker than the classical calmness condition and, hence, is more likely to hold. We perform the directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional calmness. An example is given to show that the directional quasi-normality condition may hold for the bilevel program.

Scenario Grouping and Decomposition Algorithms for Chance-Constrained Programs

A lower bound for a finite-scenario-based chance-constrained program is the quantile value corresponding to the sorted optimal objective values of scenario subproblems. This quantile bound can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. The quality of the bound depends on how the scenarios are grouped. In this paper, we formulate a mixed-integer bilevel program that optimally groups scenarios to tighten the quantile bounds. For general chance-constrained programs, we propose a branch-and-cut algorithm to optimize the bilevel program, and for chance-constrained linear programs, a mixed-integer linear-programming reformulation is derived. We also propose several heuristics for grouping similar or dissimilar scenarios. Our computational results demonstrate that optimal grouping bounds are much tighter than heuristic bounds, resulting in smaller root-node gaps and better performance of scenario decomposition for solving chance-constrained 0-1 programs. Also, the optimal grouping bounds can be greatly strengthened using larger group size. Summary of Contribution: Chance-constrained programs are in general NP-hard but widely used in practice for lowering the risk of undesirable outcomes during decision making under uncertainty. Assuming finite scenarios of uncertain parameter, chance-constrained programs can be reformulated as mixed-integer linear programs with binary variables representing whether or not the constraints are satisfied in corresponding scenarios. A useful quantile bound for solving chance-constrained programs can be improved by grouping subsets of scenarios at the expense of solving larger subproblems. In this paper, we develop algorithms for optimally and heuristically grouping scenarios to tighten the quantile bounds. We aim to improve both the computation and solution quality of a variety of chance-constrained programs formulated for different Operations Research problems.

Quadratic Program on a Structured Nonconvex Set

In this paper, we focus on a special nonconvex quadratic program whose feasible set is a structured nonconvex set. To find an effective method to solve this nonconvex program, we construct a bilevel program, where the low-level program is a convex program while the upper-level program is a small-scale nonconvex program. Utilizing some properties of the bilevel program, we propose a new algorithm to solve this special quadratic program. Finally, numerical results show that our new method is effective and efficient.

Stochastic Bilevel Program for Optimal Coordinated Energy Trading of an EV Aggregator

Gradually replacing fossil-fueled vehicles in the transport sector with Electric Vehicles (EVs) may help ensure a sustainable future. With regard to the charging electric load of EVs, optimal scheduling of EV batteries, controlled by an aggregating agent, may provide flexibility and increase system efficiency. This work proposes a stochastic bilevel optimization problem based on the Stackelberg game to create price incentives that generate optimal trading plans for an EV aggregator in day-ahead, intra-day and real-time markets. The upper level represents the profit maximizer EV aggregator who participates in three sequential markets and is called a Stackelberg leader, while the second level represents the EV owner who aims at minimizing the EV charging cost, and who is called a Stackelberg follower. This formulation determines endogenously the profit-maximizing price levels constraint by cost-minimizing EV charging plans. To solve the proposed stochastic bilevel program, the second level is replaced by its optimality conditions. The strong duality theorem is deployed to substitute the complementary slackness condition. The final model is a stochastic convex problem which can be solved efficiently to determine the global optimality. Illustrative results are reported based on a small case with two vehicles. The numerical results rely on applying the proposed methodology to a large scale fleet of 100, 500, 1000 vehicles, which provides insights into the computational tractability of the current formulation.

Convex modeling for optimal battery sizing and control of an electric variable transmission powertrain

This paper provides convex modeling steps for the problem of optimal battery sizing and energy management of a plug-in hybrid electric vehicle with an electric variable transmission. Optimal energy management is achieved by a switched model control, with driving modes identified by the engine on/off state. In pure electric mode, convex optimization is used to find the optimal torque split between two electric machines, in order to maximize powertrain efficiency. In hybrid mode, optimization is performed in a bilevel program. One level optimizes speed of a compound unit that includes the engine and electric machines. Another level optimizes the power split between the compound unit and the battery. The proposed method is used to minimize the total cost of ownership of a passenger vehicle for a daily commuter, including costs for battery, fossil fuel and electricity.

A Traffic Restriction Scheme for Enhancing Carpooling

For the purpose of alleviating traffic congestion, this paper proposes a scheme to encourage travelers to carpool by traffic restriction. By a variational inequity we describe travelers’ mode (solo driving and carpooling) and route choice under user equilibrium principle in the context of fixed demand and detect the performance of a simple network with various restriction links, restriction proportions, and carpooling costs. Then the optimal traffic restriction scheme aiming at minimal total travel cost is designed through a bilevel program and applied to a Sioux Fall network example with genetic algorithm. According to various requirements, optimal restriction regions and proportions for restricted automobiles are captured. From the results it is found that traffic restriction scheme is possible to enhance carpooling and alleviate congestion. However, higher carpooling demand is not always helpful to the whole network. The topology of network, OD demand, and carpooling cost are included in the factors influencing the performance of the traffic system.

An Evolutionary Algorithm Using Parameter Space Searching for Interval Linear Fractional Bilevel Programming Problems

Interval programming is one of main approaches treating imprecise and uncertain elements involved in optimization problems. In this paper, an interval linear fractional bilevel program is considered, which is characterized in that both objective coefficients and the right-hand side vector are interval numbers, and an evolutionary algorithm (EA) is proposed to solve the problem. First, the interval parameter space of the follower’s problem is taken as the search space of the proposed EA. For each individual, one can evaluate it by dealing with a simplified interval bilevel program in which only the leader’s objective involves interval parameters. In addition, the optimality conditions of linear fractional programs are applied to convert and solve the simplified problem. Finally, some computational examples were solved and the results show that the proposed algorithm is efficient and robust.

A Genetic Algorithm for Solving Linear-Quadratic Bilevel Program-Ming Problems

In this paper, we focus on a special linear-quadratic bilevel programming problem in which the follower’s problem is a convex-quadratic programming, whereas the leader’s functions are linear. At first, based on Karush-Kuhn-Tucher(K-K-T) conditions, the original problem is transformed into an equivalent nonlinear programming problem in which the objective and constraint functions are linear except for the complementary slack conditions. Then, a genetic algorithm is proposed to solve the equivalent problem. In the proposed algorithm, the individuals are encoded in two phases. Finally, the efficiency of the approach is demonstrated by an example.