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Constraint-solving-based program invariant synthesis takes a parametric invariant template and encodes the (inductive) invariant conditions into constraints. The problem of characterizing the set of all valid parameter assignments is…
For a quadratic matrix polynomial dependent on parameters and a given tolerance $\epsilon > 0$, the minimization of the $\epsilon$-pseudospectral abscissa over the set of permissible parameter values is discussed, with applications in…
We propose an approach to trajectory optimization for piecewise polynomial systems based on the recently proposed graphs of convex sets framework. We instantiate the framework with a convex relaxation of optimal control based on occupation…
We develop new tools in the theory of nonlinear random matrices and apply them to study the performance of the Sum of Squares (SoS) hierarchy on average-case problems. The SoS hierarchy is a powerful optimization technique that has achieved…
We prove for the first time that, if a linear inverse problem exhibits a group symmetry structure, gradient-based optimizers can be designed to exploit this structure for faster convergence rates. This theoretical finding demonstrates the…
We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements…
In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and…
With the surge of multi- and manycores, much research has focused on algorithms for mapping and scheduling on these complex platforms. Large classes of these algorithms face scalability problems. This is why diverse methods are commonly…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
Stochastic optimization algorithms update models with cheap per-iteration costs sequentially, which makes them amenable for large-scale data analysis. Such algorithms have been widely studied for structured sparse models where the sparsity…
We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated…
Disjointly constrained multilinear programming concerns the problem of maximizing a multilinear function on the product of finitely many disjoint polyhedra. While maximizing a linear function on a polytope (linear programming) is known to…
Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…
Random projection, a dimensionality reduction technique, has been found useful in recent years for reducing the size of optimization problems. In this paper, we explore the use of sparse sub-gaussian random projections to approximate…
We consider optimization problems with polynomial inequality constraints in non-commuting variables. These non-commuting variables are viewed as bounded operators on a Hilbert space whose dimension is not fixed and the associated polynomial…
In eXplainable Constraint Solving (XCS), it is common to extract a Minimal Unsatisfiable Subset (MUS) from a set of unsatisfiable constraints. This helps explain to a user why a constraint specification does not admit a solution. Finding…
Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of…
In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation…
We show that the $A$-optimal design optimization problem over $m$ design points in $\mathbb{R}^n$ is equivalent to minimizing a quadratic function plus a group lasso sparsity inducing term over $n\times m$ real matrices. This observation…
Group invariants are used in high energy physics to define quantum field theory interactions. In this paper, we are presenting the parallel algebraic computation of special invariants called symplectic and even focusing on one particular…