Power systems modeling and planning has long leveraged mathematical programming for its ability to provide optimality and feasibility guarantees. One feature that has been recognized in the optimization literature since the 1970s is the existence and meaning of multiple exact optimal and near-optimal solutions, which we call alternative solutions. In power systems modeling, the use … Read more
Consensus optimization enables autonomous agents to solve joint tasks through peer-to-peer exchanges alone. Classical decentralized gradient descent is appealing for its minimal state but fails to achieve exact consensus with fixed stepsizes unless additional trackers or dual variables are introduced. We revisit penalty methods and introduce a decentralized two-layer framework that couples an outer penalty-continuation … Read more
We consider bilevel programs that involve a leader, who first commits to a mixed-integer decision, and a follower, who observes this decision and then responds rationally by solving a linear program (LP). Standard approaches often reformulate these bilevel optimization problems as single-level mixed-integer programs by exploiting the follower’s LP optimality conditions. These reformulations introduce either … Read more
Benders’ decomposition (BD) is a framework for solving optimization problems by removing some variables and modeling their contribution to the original problem via so-called Benders cuts. While many advanced optimization techniques can be applied in a BD framework, one central technique has not been applied systematically in BD: symmetry handling. The main reason for this … Read more
We study the quadratic penalty method (QPM) for smooth nonconvex optimization problems with equality constraints. Assuming the constraint violation satisfies the PL condition near the feasible set, we derive sharper worst-case complexity bounds for obtaining approximate first-order KKT points. When the objective and constraints are twice continuously differentiable, we show that QPM equipped with a … Read more
We study the supportedness of nondominated points of multiobjective optimization problems, that is, whether they can be obtained via weighted sum scalarization. One key question is how supported points behave under an efficiency-preserving transformation of the original problem. Under a differentiability assumption, we characterize the transformations that preserve both efficiency and supportedness as the component-wise … Read more
The Low Order-Value Optimization (LOVO) problem involves minimizing the minimum among a finite number of function values within a feasible set. LOVO has several practical applications such as robust parameter estimation, protein alignment, portfolio optimization, among others. In this work, we are interested in the constrained nonlinear optimization LOVO problem of minimizing the minimum between … Read more
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Recently robust combinatorial optimization problems with budgeted interdiction uncertainty affecting cardinality-based constraints or objective were considered by presenting, comparing and experimenting with compact formulations. In this paper we present a compact formulation for the case in which locally budgeted interdiction uncertainty affects the objective function and covering constraints simultaneously. ArticleDownload View PDF
The SCIP Optimization Suite provides a collection of software packages for mathematical optimization, centered around the constraint integer programming (CIP) framework SCIP. This report discusses the enhancements and extensions included in SCIP Optimization Suite 10.0. The updates in SCIP 10.0 include a new solving mode for exactly solving rational mixed-integer linear programs, a new presolver … Read more