DDMA works on some of the most challenging problems in optimization. Objective functions are often high-dimensional, computationally expensive, and come with complex nonlinear constraints. Derivative information may be incomplete or missing. Sometimes problems are best expressed using categorical or noncontinuous variables. Many applications are time-sensitive, requiring an acceptable solution within a fixed time.

Research:

• Reducing runtime via cpu-time control. Objective functions that are expensive (in time) to evaluate often limit the usefulness of optimzation in time-sensitive applications. Straightforward ways to address this difficulty are parallelization and increased computational power. We are exploring methods for reducing unrofitable CPU burden through the use of surrogate functions and objective function control via subprocess CPU-time monitoring. see more
• Optimization on manifolds. Equality constraints are common and often difficult to enforce. They can be treated efficiently as searches on mainfolds without the use of filter methods. We are exploring and developing direct search methods applied to manifolds. see more
• Finding Computationally Tractable Methods. Optimization problems with high worst-case computational complexity often have very modest typical-case complexity. Characterizing the statistical ensemble of such problems leads to an algorithmic phase structure 2 that can predict when such methods will be computationally tractable.
• Mixed-variable programming and direct-search methods. MVP methods are employed when some optimization variables are discrete (e.g integer), categorical (e.g. color, material), or when the problem dimension is variable. These methods are often successfully applied to problems otherwise considered too difficult because of the unusual variable constraints. Direct search methods are employed when derivative information is unavailable or unreliable. see more
• Regularized functional minimization. Some optimization problems seek to minimize a measure on a class of functions. This situation is ubiquitous in signal and image analysis, and in image restoration tasks. Various methods are used when a functional derivative exists. see more
• Multi-objective and filter methods. Some problems are often best formulated in terms of multiple objectives. This can arise from multiple independent measures of solution quality or from a relaxed representation of constraints. Filter methods seek to improve optimization efficiency through controlled constraint violation, and they can also be used to mollify the effects of noisy objectives. see more