Is is possible to "delete" a memory permanently? If cognitive memory is compared to computer memory, then is it possible to purposefully and permanently forget a specific memory, using known technologies and methodologies?
There is still much that is not completely understood about the process of memory and forgetfulness, and the following is what I have been able to find out.
It is understoood that physiologically, the establishment of long-term memory involves a process of physical changes in the structure of neurons (or nerve cells) and their synapses in the brain, through a process known as long-term potentiation (LTP). (Mastin, 2018; Science Direct, 2018a).
Short-term memory can become long-term memory through the process of consolidation, involving rehearsal and meaningful association (strengthening the physical structure of neurons and their synapses). Unlike short-term memory (which relies mostly on an acoustic, and to a lesser extent a visual, code for storing information), long-term memory encodes information for storage semantically (i.e. based on meaning and association).
The opposite to LTP is long-term depression (LTD) (Dharani, 2015), whereby the neuronal and synaptic structures are weakened, and forgetfulness develops, after a certain triggering activity. (Science Direct, 2018b).
There are several forms of LTP and LTD classified by their mechanisms and their mode of induction (Hammond, et al. 2015), and LTD can be temporary or permanent.

LTD synergistically functions with long-term potentiation (LTP) to generate a balanced, stable memory mechanism. Although LTP plays a major role in the associative type of learning that occurs in the hippocampus and neocortex, LTD is a major player in the error-driven learning that occurs in the cerebellar cortex. Recent studies have revealed complex signal transduction processes underlying LTD. In the long run, LTD may lead to a loss of synapses, or to preservation as silent synapses that have recently been found to prevail in the cerebellar cortex (Ito, 2009).

To answer your question

is it possible to purposefully and permanently forget a specific memory

It seems possible, and Hammond, et al. (2015) discusses induction of LTD, however induction of LTD to forget a specific memory would be impossible at this present time that I can find.
References
Dharani, K. (2015). Chapter 3 - Memory. The Biology of Thought: A Neuronal Mechanism in the Generation of Thought–A New Molecular Model p.53-74. doi: 10.1016/B978-0-12-800900-0.00003-8
Hammond, C., Goaillard, J-M., Debanne, D. &amp; Gaiarsa, J-L. (2015). Chapter 18 - Synaptic Plasticity. Cellular and Molecular Neurophysiology (Fourth edition) p361-389. doi: 10.1016/B978-0-12-397032-9.00018-2
Ito, M. (2009). Long-Term Depression: Cerebellum. Encyclopedia of Neuroscience, p533-539. doi: 10.1016/B978-008045046-9.00807-X
Mastin, L. (2018). Long-Term Memory [Online]Retreived from: http://www.human-memory.net/types_long.html
Science Direct (2018a). Long Term Potentiation [Online]Retreived from: https://www.sciencedirect.com/topics/biochemistry-genetics-and-molecular-biology/long-term-potentiation
Science Direct (2018b). Long Term Depression [Online]Retreived from: https://www.sciencedirect.com/topics/biochemistry-genetics-and-molecular-biology/long-term-depression