Jong-Shi Pang’s research while affiliated with University of Southern California and other places

Indefinite quadratic programs (QPs) are known to be very difficult to be solved to global optimality, so are linear programs with linear complementarity constraints. Treating the former as a subclass of the latter, this paper presents a progressive mixed integer linear programming method for solving a general linear program with linear complementarity constraints (LPCC). Instead of solving the LPCC with a full set of integer variables expressing the complementarity conditions, the presented method solves a finite number of mixed integer subprograms by starting with a small fraction of integer variables and progressively increasing this fraction. After describing the PIP (for progressive integer programming) method and its various implementations, we demonstrate, via an extensive set of computational experiments, the superior performance of the progressive approach over the direct solution of the full-integer formulation of the LPCCs. It is also shown that the solution obtained at the termination of the PIP method is a local minimizer of the LPCC, a property that cannot be claimed by any known non-enumerative method for solving this nonconvex program. In all the experiments, the PIP method is initiated at a feasible solution of the LPCC obtained from a nonlinear programming solver, and with high likelihood, can successfully improve it. Thus, the PIP method can improve a stationary solution of an indefinite QP, something that is not likely to be achievable by a nonlinear programming method. Finally, some analysis is presented that provides a better understanding of the roles of the LPCC suboptimal solutions in the local optimality of the indefinite QP.

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].

There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being discontinuous and a minimizer (local or global) of the problems, even if it exists, being impossible to verifiably compute, a foremost question is what kind of ''stationary solutions'' one can expect to obtain; these solutions provide promising candidates for minimizers; i.e., their defining conditions are necessary for optimality. Motivated by recent results on sparse optimization, we introduce in this paper such a kind of solution, termed ''pseudo B- (for Bouligand) stationary solution'', for a broad class of discontinuous piecewise continuous optimization problems with objective and constraint defined by indicator functions of the positive real axis composite with functions that are possibly nonsmooth. We present two approaches for computing such a solution. One approach is based on lifting the problem to a higher dimension via the epigraphical formulation of the indicator functions; this requires the addition of some auxiliary variables. The other approach is based on certain continuous (albeit not necessarily differentiable) piecewise approximations of the indicator functions and the convergence to a pseudo B-stationary solution of the original problem is established. The conditions for convergence are discussed and illustrated by an example.

This paper studies several solution paths of sparse quadratic minimization problems as a function of the weighing parameter of the bi-objective of estimation loss versus solution sparsity. Three such paths are considered: the “ $\ell _0$ ℓ 0 -path” where the discontinuous $\ell _0$ ℓ 0 -function provides the exact sparsity count; the “ $\ell _1$ ℓ 1 -path” where the $\ell _1$ ℓ 1 -function provides a convex surrogate of sparsity count; and the “capped $\ell _1$ ℓ 1 -path” where the nonconvex nondifferentiable capped $\ell _1$ ℓ 1 -function aims to enhance the $\ell _1$ ℓ 1 -approximation. Serving different purposes, each of these three formulations is different from each other, both analytically and computationally. Our results deepen the understanding of (old and new) properties of the associated paths, highlight the pros, cons, and tradeoffs of these sparse optimization models, and provide numerical evidence to support the practical superiority of the capped $\ell _1$ ℓ 1 -path. Our study of the capped $\ell _1$ ℓ 1 -path is interesting in its own right as the path pertains to computable directionally stationary (= strongly locally minimizing in this context, as opposed to globally optimal) solutions of a parametric nonconvex nondifferentiable optimization problem. Motivated by classical parametric quadratic programming theory and reinforced by modern statistical learning studies, both casting an exponential perspective in fully describing such solution paths, we also aim to address the question of whether some of them can be fully traced in strongly polynomial time in the problem dimensions. A major conclusion of this paper is that a path of directional stationary solutions of the capped $\ell _1$ ℓ 1 -regularized problem offers interesting theoretical properties and practical compromise between the $\ell _0$ ℓ 0 -path and the $\ell _1$ ℓ 1 -path. Indeed, while the $\ell _0$ ℓ 0 -path is computationally prohibitive and greatly handicapped by the repeated solution of mixed-integer nonlinear programs, the quality of $\ell _1$ ℓ 1 -path, in terms of the two criteria—loss and sparsity—in the estimation objective, is inferior to the capped $\ell _1$ ℓ 1 -path; the latter can be obtained efficiently by a combination of a parametric pivoting-like scheme supplemented by an algorithm that takes advantage of the Z-matrix structure of the loss function.

The problem of inferring Markov random fields (MRFs) with a sparsity or robustness prior can be naturally modeled as a mixed-integer program. This motivates us to study a general class of convex submodular optimization problems with indicator variables, which we show to be polynomially solvable in this paper. The key insight is that, possibly after a suitable reformulation, indicator constraints preserve submodularity. Fast computations of the associated Lov\'asz extensions are also discussed under certain smoothness conditions, and can be implemented using only linear-algebraic operations in the case of quadratic objectives.